Solving Quadratic Equation by Formula(更正)
c语言解方程式
c语言解方程式In the realm of computer programming, C language stands as a powerful tool for numerical computations, including solving equations. When it comes to solving equations in C, there are several approaches that can be employed, depending on the nature and complexity of the equation.在计算机编程领域，C语言是一个用于数值计算（包括解方程式）的强大工具。
在C语言中解方程式时，可以根据方程式的性质和复杂度采用不同的方法。
For simple algebraic equations, such as linear or quadratic equations, we can implement algorithms that directly apply the relevant formulas to find the solution. For example, to solve a quadratic equation ax²+ bx + c = 0, we can use the quadratic formula x = [-b ± √(b² - 4ac)] / (2a). In C, we would calculate the discriminant, determine the sign of the root, and then compute the solutions accordingly.对于简单的代数方程，如线性方程或二次方程，我们可以实现直接应用相关公式的算法来找到解。
二元一次方程的顶点公式
二元一次方程的顶点公式A quadratic equation is a mathematical expression of the form ax² + bx + c = 0, where a, b, and c are constants and x is the variable. When graphed, a quadratic equation forms a parabolic curve, and the vertex of this parabola is a crucial point on the curve. Finding the vertex of a quadratic equation is essential for understanding its behavior and making predictions based on its graph.二元一次方程是一个数学表达式，形式为ax² + bx + c = 0，其中a、b和c是常数，x是变量。
当作图时，二次方程形成一个抛物线曲线，而这个抛物线的顶点是曲线上的一个关键点。
找到二次方程的顶点是理解其行为并根据其图表做出预测的关键。
The vertex of a quadratic equation can be calculated using the formula x = -b/2a. This formula provides the x-coordinate of the vertex, which can then be substituted back into the original equation to find the y-coordinate. By determining the vertex of a quadratic equation, we can determine the equation's minimum or maximum value, depending on whether the parabola opens upwards or downwards.可以使用公式x = -b/2a来计算二次方程的顶点。
卡丹公式推导全过程
卡丹公式推导全过程The Cardano formula, also known as Cardano's method, is a method for solving cubic equations. This formula was derived by the Italian mathematician Gerolamo Cardano in his 1545 book Ars Magna. The Cardano formula is an important and fundamental result in mathematics, as it provides a way to find the roots of cubic equations using only elementary algebraic operations.卡丹公式，也被称为卡丹方法，是一种解决三次方程的方法。
这个公式是由意大利数学家杰罗拉莫·卡丹在他的1545年著作《Ars Magna》中导出的。
卡丹公式是数学中的一个重要而基本的结果，因为它提供了一种只使用基本代数运算就能找到三次方程的根的方法。
The Cardano formula can be applied to any cubic equation of the form \(ax^3 + bx^2 + cx + d = 0\), where \(a \neq 0\). The first stepin using the Cardano formula is to remove the quadratic term by making a substitution \(x = y - \frac{b}{3a}\). This simplifies the equation to a depressed cubic form, \(y^3 + py + q = 0\), where \(p = \frac{3ac - b^2}{3a^2}\) and \(q = \frac{2b^3 - 9abc +27a^2d}{27a^3}\).卡丹公式可以应用于任何形式为\(ax^3 + bx^2 + cx + d = 0\)的三次方程，其中\(a \neq 0\)。
Solve Quadratic Equations using Factoring
Slide 6.5 - 8
Solve quadratic equations by factoring. (cont’d)
In summary, follow these steps to solve quadratic equations by factoring.
Step 1: Write the equation in standard form— that is, with all terms on one side of the equals sign in descending power of the variable and 0 on the other side.
Slide 6.5 - 2
Objective
Solve quadratic equations by factoring.
Solve quadratic equations by factoring.
The diagram above suggests the following key points: One side of the equation is just zero. The opposite side should contain the factors of the
Slide 6.5 - 6
EXAMPLE 1 Using the Zero-Factor Property
Solve.
Solution:
2x 35x 7 0 2x 3 0 or 5x 7 0
3 2
,
7 5
2x 33 0 3 5x 7 7 0 7
2x 3 22
5x 7 55
Solving Quadratic Equations by Factoring
三角函数公式大全（Trigfunctionformula）
三角函数公式大全（Trig function formula）Induced formulaSin (-a) =-sin (a)Cos (-a) =cos (a)Sin (pi/2-a) =cos (a)Cos (pi/2-a) =sin (a)Sin (pi/2+a) =cos (a)Cos (pi/2+a) =-sin (a)Sin (pi-A) =sin (a)Cos (pi-A) =-cos (a)Sin (pi+a) =-sin (a)Cos (pi+a) =-cos (a)TgA=tanA=sinA/cosAThe horns and trigonometric function and difference Sin (a+b), =sin (a), cos (b), +cos (alpha), sin (b) Cos (a+b), =cos (a), cos (b), -sin (a), sin (b)Sin (a-b), =sin (a), cos (b), -cos (a), sin (b)Cos (a-b), =cos (a), cos (b), +sin (a), sin (b)Tan (a+b) = (Tan (a) +tan (b)) / (1-tan (a) Tan (b))Tan (a-b) = (Tan (a) -tan (b)) / (1+tan (a) Tan (b))Sum difference product formulaSin (a), +sin (b), =2sin ((a+b) /2), cos ((a-b) /2)Sin (a), sin (b), =2cos ((a+b) /2), sin ((a-b) /2)Cos (a), +cos (b), =2cos ((a+b) /2), cos ((a-b) /2)Cos (a), -cos (b), =-2sin ((a+b) /2), sin ((a-b) /2) Integral and difference formulaSin (a), sin (b), =-1/2*[cos (a+b), -cos (a-b)]Cos (a), cos (b), =1/2*[cos (a+b), +cos (a-b)]Sin (a), cos (b), =1/2*[sin (a+b), +sin (a-b)]Two fold angle formulaSin (2a), =2sin (a), cos (a)Cos (2a), =cos^2 (a), -sin^2 (a), =2cos^2 (a), -1=1-2sin^2 (a)Half angle formulaSin^2 (a/2) = (1-cos (a)) /2Cos^2 (a/2) = (1+cos (a)) /2Tan (a/2) = (1-cos (a)) /sin (a) =sin (a) / (1+cos (a))Universal formulaSin (a) = (2tan (a/2)) / (1+tan^2 (a/2))Cos (a) = (1-tan^2 (a/2)) / (1+tan^2 (a/2))Tan (a) = (2tan (a/2)) / (1-tan^2 (a/2))Other formulasA*sin (a), +b*cos (a), =sqrt (a^2+b^2), sin (a+c) [where, Tan (c), =b/a]A*sin (a), -b*cos (a), =sqrt (a^2+b^2), cos (A-C) [where, Tan (c), =a/b]1+sin (a) = (sin (a/2) +cos (a/2)) ^21-sin (a) = (sin (a/2) -cos (a/2)) ^2Other non focusCSC (a) =1/sin (a)Sec (a) =1/cos (a)Hyperbolic functionSinh (a) = (e^a-e^ (-a)) /2Cosh (a) = (e^a+e^ (-a)) /2TGH (a), =sinh (a), /cosh (a)Multiplication and factorization, a2-b2= (a+b) (a-b), a3+b3= (a+b) (a2-ab+b2), a3-b3= (a-b) (a2+ab+b2)The triangle inequality is less than or equal to |a|+|b| less than or equal to |a|+|b| |a-b| |a+b| |a| = b<=>-b = a = b|a-b| = |a|-|b| -|a| = a = |a|-b+ have the solutions of quadratic equation with one unknown of the (b2-4ac) /2a -b-b+ (b2-4ac) /2a VRelation between roots and coefficients X1+X2=-b/a, X1*X2=c/a notes: the laws of WeberDiscriminant b2-4a=0 note: equations have equal two real rootsB2-4ac>0 note: the equation has a real rootB2-4ac<0 note: the equation has conjugate complex rootsTrigonometric function formulaThe horns and the formula sin (A+B) =sinAcosB+cosAsinB sin (A-B) =sinAcosB-sinBcosACos (A+B) =cosAcosB-sinAsinB cos (A-B) =cosAcosB+sinAsinBTan (A+B) = (tanA+tanB) / (1-tanAtanB) Tan (A-B) = (tanA-tanB) / (1+tanAtanB)CTG (A+B) = (ctgActgB-1) / (ctgB+ctgA) CTG (A-B) = (ctgActgB+1) / (ctgB-ctgA)Double angular formula tan2A=2tanA/ (1-tan2A) ctg2A= (ctg2A-1) /2ctgaCos2a=cos2a-sin2a=2cos2a-1=1-2sin2aHalf angle formula sin (A/2) = V ((1-cosA) /2) sin (A/2) = V ((1-cosA) /2)Cos (A/2) = V ((1+cosA) /2) cos (A/2) = V ((1+cosA) /2)Tan (A/2) = V ((1-cosA) / ((1+cosA)) Tan (A/2) = V ((1-cosA) / ((1+cosA))CTG (A/2) = V ((1+cosA) / ((1-cosA)) CTG (A/2) = V ((1+cosA) / ((1-cosA))Sum and difference product, 2sinAcosB=sin (A+B), +sin (A-B),2cosAsinB=sin (A+B), -sin (A-B)2cosAcosB=cos (A+B), -sin (A-B), -2sinAsinB=cos (A+B), -cos (A-B)SinA+sinB=2sin ((A+B) /2) cos ((A-B) /2 cosA+cosB=2cos ((A+B) /2) sin ((A-B) /2)TanA+tanB=sin (A+B) /cosAcosB tanA-tanB=sin (A-B) /cosAcosBCtgA+ctgBsin (A+B) /sinAsinB -ctgA+ctgBsin (A+B) /sinAsinBCertain sequences, first n items, and 1+2+3+4+5+6+7+8+9+... +n=n (n+1) /2 1+3+5+7+9+11+13+15+... + (2n-1) =n22+4+6+8+10+12+14+... + (2n) =n (n+1)12+22+32+42+52+62+72+82+... +n2=n (n+1) (2n+1) /613+23+33+43+53+63+... N3=n2 (n+1) 2/41*2+2*3+3*4+4*5+5*6+6*7+... +n (n+1), =n (n+1) (n+2), /3Note: the sine theorem of a/sinA=b/sinB=c/sinC=2R where R is the radius of circumcircle of triangleCosine theorem b2=a2+c2-2accosB note: the angle B is the angle between the edge a and the edge CThe standard equation of the circle (x-a), 2+ (y-b) 2=r2 note: (a, b) is the coordinate of the center of the circleGeneral equation of circle x2+y2+Dx+Ey+F=0 note: D2+E2-4F>0Parabolic standard equation y2=2px, y2=-2px, x2=2py, x2=-2pyStraight prism side area S=c*h oblique prism side area S=c'*h Orthoprism side area S=1/2c*h'is bevel side area (c+c') H'S=1/2The surface area of the frustum of the cone is S=1/2 (c+c') l=pi (R+r) l, and the surface area of the ball is S=4pi*r2The cylinder side area is S=c*h=2pi*h, and the cone side area is S=1/2*c*l=pi*r*lThe arc length formula l=a*r, a is the radian number of the center angle, R, >0 fan area formula s=1/2*l*rCone volume formula V=1/3*S*H cone volume formula V=1/3*pi*r2hNote: the oblique prism volume V=S'L, S'is a straight section area, L is the long side edgeCylindrical volume formula V=s*h cylinder V=pi*r2h。
SAT数学知识点
SAT 数学知识点一Number and Operations Review 一、Properties of integers知道下列说法表示的内容：1. Integers consist of the whole numbers and their negatives (including zero).2. Integers extend infinitely in both negative and positive directions.3. Integers do not include fractions or decimals.4. Negative integers5. Positive integers6. The integer zero is neither positive nor negative.7. odd numbers（奇数）and even numbers（偶数）8. Consecutive integers9. Addition of integers(奇数偶数的加法规则)10. Multiplication of integers（奇数偶数的乘法规则）二、Arithmetic word problems（算术题）三、Number lines（数轴）四、Square and square roots（平方和平方根）五、Fractions and rational numbers（分数与有理数）六、Elementary number theory☆Factors, multiples, and remainders☆Prime numbers七、Ratios, proportions, and percents八、Sequences九、Sets(union, intersection, elements)十、Counting problems Counting problems involve figuring out how many ways you can select or arrange members of groups, such as letters of the alphabet, numbers or menu selections.☆Fundamental counting problems分步完成事件和分类完成事件发生的可能性☆Permutations and combinations （排列组合）基本排列组合理论十一、Logical reasoningThe SAT doesn’t include1.Tedious or long computations2.Matrix operations1.2.3.4.5. 6.7.8.9.10.11.12.13. 14.SAT数学知识点二Algebra and Functions Review Many math questions require knowledge of algebra. This chapter gives you some further practice. You have to manipulate and solve a simple equation for an unknown, simplify and evaluate algebraic expressions, and use algebraic expressions, and use algebraic concepts in problem-solving situations.For the math questions covering algebra and functions content, you should be familiar with all of the following basic skills and topics:一、Operations on algebraic expressions二、Factoring三、Exponents四、Evaluating expressions with exponents and roots五、Solving equations☆Working with “unsolvable” equations☆Solving for one variable in terms of another☆Solving equations involving radical expressions六、Absolute value 七、Direct translation into mathematical expressions八、Inequalities九、Systems of linear equations and inequalities十、Solving quadratic equations by factoring 十一、Rational equations and inequalities 十二、Direct and inverse variation十三、Word problems十四、Functions☆Function notation and evaluation☆Domain and range☆Using new definitions☆Functions as models☆Linear functions: their equations and graphs☆Quadratic functions: their equations and graphs☆Qualitative behavior of graphs and functions☆Translations and their effects on graphsand functionsThe SAT doesn’t include:一、Solving quadratic equations thatrequire the use of the quadraticformula二、Complex numbers三、Logarithms1.2.3.4. 5.6.7.8.9.10.SAT 数学知识点三Geometry and Measurement Review Concept you should to knowFor the mathematics questions covering geometry and measurement concepts, you should be familiar with all of the following basic skills, topics, and formulas:一、Geometric notation二、Points and lines三、Angles in the plane四、Triangles(including special triangles)☆Equilateral triangles☆Isosceles triangles☆Right triangles and the Pythagorean theorem ☆30º-60º-90ºtriangles☆45º-45º-90ºtriangles☆3-4-5 triangles☆Congruent triangles☆Similar triangles☆The triangle inequality五、Quadrilaterals☆Parallelograms☆Rectangles☆Squares六、Areas and Perimeters☆Areas of squares and rectangles☆Perimeters of squares and rectangles☆Area of triangles☆Area of Parallelograms七、Other polygons☆Angles in a polygon☆Perimeter☆Area八、Circles☆Diameter☆Radius☆Arc☆Tangent to a circle☆Circumference☆Area九、Solid geometry☆Solid figures and volumes☆Surface area十、Geometric perception十一、Coordinate geometry☆Slopes, parallel lines, and perpendicular lines☆The midpoint formula☆The distance formula十二、TransformationsThe SAT doesn’t include:一、Formal geometric proofs二、Trigonometry三、Radian measure1.2.3.4.5.6. 7.8.9.SAT 数学知识点四Data Analysis, Statistics andProbability ReviewFor the math questions covering data analysis, statistics and probability concepts, you should be familiar with all of the following basic skills and topics:一、Data interpretation二、Statistics☆Arithmetic mean☆Median☆Mode☆Weighted average☆Average of algebraic expression☆Using average to find missing numbers三、Elementary probability四、Geometric probabilityThe SAT doesn’t include:四、Computation of standard deviation 1.2.3.4.5. 6.7.8.Word Problems1.2.3.4. 5-75.6.7.1112。
Quadratic Equations
To solve a quadratic equation we get it in the form above and see if it will factor.
x 5x 6
2
Get form above by subtracting 5x and adding 6 to both sides to get 0 on right side.
x2 6x 9 6
Now factor the left hand side.
x 3x 3 6
two identical factors
This can be written as:
x 3
2
6
Now we'll get rid of the square by square rooting both sides. Remember you need both the positive and negative root! Subtract 3 from both sides to get x alone. These are the answers in exact form. We can put them in a calculator to get two approximate answers.
2 x 3x 0
2
Factor out the common x Use the Null Factor law and set each factor = 0 and solve.
x2 x 3 0
x 0 or 2 x 3 0
3 x 0 or x 2
If you put either of these values in for x in the original equation you can see it makes a true statement.
高中必修数学知识点总结及公式大全
高中必修数学知识点总结及公式大全1.二次函数的标准形式为y=ax^2+bx+c。
The standard form of a quadratic function is y=ax^2+bx+c.2.一次函数的标准形式为y=kx+b。
The standard form of a linear function is y=kx+b.3.三角函数sin、cos、tan分别表示正弦、余弦、正切。
The trigonometric functions sin, cos, tan represent sine, cosine, tangent respectively.4.三角函数的周期性是它们的重要特征之一。
The periodicity of trigonometric functions is one oftheir important characteristics.5.平行四边形的面积公式为S=底×高。
The formula for the area of a parallelogram isS=base×height.6.直角三角形的勾股定理为a^2 + b^2 = c^2。
The Pythagorean theorem for a right-angled triangle isa^2 + b^2 = c^2.7.两点间距离公式为d=sqrt[(x2-x1)^2 + (y2-y1)^2]。
The distance formula between two points is d=sqrt[(x2-x1)^2 + (y2-y1)^2].8.二次方程的解法包括用公式法和配方法。
The methods for solving quadratic equations include using the formula and completing the square.9.函数奇偶性的判定方法是f(-x) = f(x)或f(-x) = -f(x)。
完全平方公式一等奖课件
完全平方公式一等奖课件Title: The First Prize Courseware on the Quadratic FormulaIntroduction:I. What is a Perfect Square Trinomial?1. Definition: A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial.2. Structure: A perfect square trinomial follows the general form: (ax + b)^2 = a^2x^2 + 2abx + b^2.。
3. Characteristic Features: Key features include a perfect square binomial and positive leading coefficient.II. Applying the Perfect Square Formula:1. Derivation: The Quadratic Formula can be derived by expanding and factoring a perfect square trinomial.2. Constant Coefficient: The constant coefficient of a perfect square trinomial is always the square of half the coefficient of the linear term.3. Square of a Binomial: The perfect square trinomial can be factored into (a + b)^2, where a represents the root(s) of the quadratic equation.III. Solving Quadratic Equations:1. General Form: A quadratic equation can be represented as ax^2 + bx + c = 0, where a, b, and c are constants.2. Discerning Perfect Square Trinomials: Recognizing the perfect square form is crucial in the factorization of quadratic equations.3. Solving Using Quadratic Formula: Apply the Quadratic Formula: x = (-b ± √(b^2 - 4ac)) / (2a) to find the roots.IV. Real-Life Applications:1. Architecture: The construction of arches and parabolic structures often employs quadratic principles.2. Physics: Motion problems involving projectiles, gravity, and vertical displacement can be modeled using quadratic equations.3. Finance: Calculating profit and loss, as well as determining break-even points, can be done through quadratic equations.V. Example Problems:1. Factorization: Given a quadratic equation, factorize it into a perfect square trinomial.2. Finding Roots: Find the roots of a given quadratic equation using the Quadratic Formula.3. Word Problems: Solve real-life word problems involving quadratic equations.Conclusion:In conclusion, we have explored the concept of perfect square trinomials and the Quadratic Formula in this first prize courseware. We have learned how to recognize and apply the perfect square form, solve quadratic equations using the Quadratic Formula, and examined real-life applications. Mastering the Quadratic Formula will equip you with the skills to solve various mathematical and practical problems. Thank you for your participation, and I hope you found this courseware both informative and engaging!。
Algebra1中英版对照目录
分步和混合运算解一元一次方程 P92—P99 2—4：Solving Equations with Variables on Both Sides
变量在两边的一元一次方程的解法 P100—P106 2—5：Solving for a Variable 求变量的值（一元一次）P107—P113
指数幂的乘法运算（同底数幂相乘和积的乘方运算）P460—P466 7—4：Division Properties of Exponents 指数幂的除法 P467—P474 Quiz for Lessons 7-1 through 7-4 第七章 1—4 课小测试 P474—P475 7—5：Polynomials 多项式 P476—P483 7—6：Adding and Subtracting Polynomials 多项式的加减运算 P484—P491 7—7：Multiplying Polynomials 多项式的乘法（单项式与多项式相乘、多项
Algebra 1 与中文数学课本内容在顺序上的对照
Algebra 1
中教数学
CHAPTER 0---- To The Student 0—1：Geometry Formulas
几何公式（s quare，rec tangle,triangle,c irc le）Z3— Z4 0—2：Tree Diagrams 树状图 Z4—Z6 0—3：The Coordinate Plane 直角坐标系 Z7—Z8 0—4：Rounding and Estimating 四舍五入 Z9—Z11 0—5：Adding and Subtracting Decimals 小数的加减运算 Z12—Z13 0—6：Multiplying and Dividing Decimals 小数的乘除运算 Z14—Z16 0—7：Prime and Composite Numbers 素数与合数 Z17—Z18 0—8：Factoring 因数 Z19—Z20 0—9：GCF and LCM 最大公约数和最小公倍数 Z21—Z22
solve-SolveEquations
solve - Solve EquationsCalling Sequence:solve(eqn, var)solve(eqns, vars)Parameters:eqn - an equation, inequality or procedureeqns - a set of equations or inequalitiesvar - (optional) a name (unknown to solve for)vars - (optional) a set of names (unknowns to solve for)Description:•The solution to a single equation eqn solved for a single unknown var is returned as an expression. To solve a system of equations for some unknowns the system is specified as a set of equations eqns and a set of unknowns vars. The solutions for vars are returned as a set of equations.•The output from solve in general is a sequence of solutions. In the case of multiple solutions, you should put the solutions in a list or set before manipulating them further. When solve is unable to find anysolutions, the empty sequence NULL is returned. This means that either there are no solutions, or simply that solve was unable to find the solutions. If solve was unable to find some solutions, it will set the global variable _SolutionsMayBeLost to true.•Two shortcuts for the input of equations and unknowns are provided. Wherever an equation is expected, if an expression expr is specified then the equation expr = 0 is understood. If vars is not specified, Maple will solve for all the variables that appear in eqns, i.e. indets(eqns,name) is used in place of vars.•In general, explicit solutions in terms of radicals for polynomial equations of degree greater than 4 do not exist. In these cases, implicit solutions are given in terms of RootOfs. Note that explicit solutions for the general quartic polynomial equation are not by default given because they are too complicated and messy to be of use. By setting the global variable _EnvExplicit to true then solve will return explicit solutions for quartic polynomials in all cases. By setting _EnvExplicit to false, all solutions which are not rational will be reported as RootOfs.•The number of solutions found can be controlled by changing the value of the global variable _MaxSols.If an integer is assigned to _MaxSols, only those many solutions will be computed and returned. Thevariable _EnvAllSolutions, if set to true, will force all inverse transcendental functions to return the entire set of solutions. This usually requires additional, system created, variables, which take integer values.Normally such variables are named with prefix _Z for integer values, _NN for non-negative integervalues and _B for binary values (0 and 1).•A single quadratic equation with constant coefficients in one variable is solved directly by substitution into the quadratic formula. However, if _EnvTryHard is set to true, Maple will try to express solutions in the common radical basis. This may lead to a nicer answer, but can take a long time. •When abs is used in the context of solve, solve assumes that the argument to abs is real-valued.•For solving differential equations, use dsolve . For purely floating-point solutions use fsolve . Use isolve to solve for integer solutions, msolve to solve modulo a prime; rsolve for recurrences, and linalg[linsolve] to solve matrix equations. •Further information is available for the subtopics solve[subtopic] where subtopic is one offloat function identity ineq linear radical scalar series system•For systems of polynomial equations, the function Groebner[gsolve] which uses a Grobner-basis approach may be useful.Examples:> solve( f=m*a, a );f m> solve( {f=m*a}, {a} );{}=a f m > f := proc(x) x-cos(x) end proc:solve( f(x),x);()RootOf − _Z ()cos _Z > eq := x^4-5*x^2+6*x=2;:= eq = − + x 45x 26x 2> solve(eq,x);,,,111−1> sols := [solve(eq,x)];:= sols [,,,111−1> sols[1];1> evalf(sols);[],,,1.1..732050808-2.732050808> sols := {solve(eq,x)};:= sols {},1−11> sols[1];1> subs( x=sols[1], eq );= 22> solve(x^4+x+1,x);()RootOf , + + _Z 4_Z 1 = index 1()RootOf , + + _Z 4_Z 1 = index 2,,()RootOf , + + _Z 4_Z 1 = index 3()RootOf , + + _Z 4_Z 1 = index 4,> evalf({%});− .7271360845.9340992895I − -.7271360845.4300142883I ,,{ + -.7271360845.4300142883I + .7271360845.9340992895I ,}Manipulating Solutions: solve for u,v,w> eqns := {u+v+w=1, 3*u+v=3, u-2*v-w=0};:= eqns {},, = + 3u v 3 = − − u 2v w 0 = + + u v w 1> sols := solve( eqns );:= sols {},, = v 35 = w -25 = u 45check solutions> subs( sols, eqns );{},, = 33 = 11 = 00pick off one solution> subs( sols, u );45assign all solutions> assign( sols );u;45Other Examples> solve( cos(x)+y = 9, x );− π()arccos − y 9> solve( x^2+x>5, x );, RealRange ,−∞ Open − − 12RealRange , Open − + 12∞> solve( x^6-2*x^2+2*x, x );0()RootOf , − + _Z 52_Z 2 = index 1()RootOf , − + _Z 52_Z 2 = index 2,,,()RootOf , − + _Z 52_Z 2 = index 3()RootOf , − + _Z 52_Z 2 = index 4,,()RootOf , − + _Z 52_Z 2 = index 5> solve( {x^2*y^2=0, x-y=1} );,,,{}, = y 0 = x 1{}, = y 0 = x 1{}, = x 0 = y -1{}, = x 0 = y -1> solve( {x^2*y^2=0, x-y=1, x<>0} );,{}, = y 0 = x 1{}, = y 0 = x 1> solve( {x^2*y^2-b, x^2-y^2-a}, {x,y} );{}, = y ()RootOf + − _Z 4_Z 2a b = x ()RootOf − − _Z 2()RootOf + − _Z 4_Z 2a b 2a> _EnvAllSolutions := true:solve( sin(x)=cos(x)-1, x );,−+ 12π2π_Z12π_Z2> r := solve(cos(x),x);:= r + 12ππ_Z3~> subs(op(indets(r))=3, r); # substitute 3 for _Z3~ in r72πSee Also:RootOf , allvalues , dsolve , fsolve , isolve , msolve , rsolve , assign , invfunc , isolate , match , linalg[linsolve], simplex , Groebner。
英语作文四宫格写作法
英语作文四宫格写作法Quadratic Formula.The quadratic formula is a formula that can be used to find the solutions to a quadratic equation. A quadratic equation is any equation that can be written in the form`ax^2 + bx + c = 0`, where `a`, `b`, and `c` are real numbers and `a` is not equal to 0. The quadratic formula is:x = (-b ± √(b² 4ac)) / 2a.where `a`, `b`, and `c` are the coefficients of the quadratic equation.To use the quadratic formula, follow these steps:1. Identify the coefficients of the quadratic equation: `a`, `b`, and `c`.2. Substitute the values of `a`, `b`, and `c` into thequadratic formula.3. Simplify the expression under the square root sign.4. Calculate the two values of `x`.Example:Find the solutions to the quadratic equation `x^2 + 5x + 6 = 0`.Solution:1. Identify the coefficients of the quadratic equation: `a = 1`, `b = 5`, and `c = 6`.2. Substitute the values of `a`, `b`, and `c` into the quadratic formula:x = (-5 ± √(5² 4(1)(6))) / 2(1)。
一元二次方程求根公式和最值
一元二次方程求根公式和最值Quadratic equations are a fundamental topic in mathematics that students often struggle with. The quadratic formula, also known as the "求根公式," is used to solve quadratic equations of the form ax^2 + bx + c = 0. This formula provides the roots or solutions of the equation, which are the values of x where the equation equals zero. Understanding how to apply the quadratic formula is crucial for success in algebra and beyond.一元二次方程是数学中的一个基本主题，学生们经常在这个问题上感到困难。
一元二次方程的求根公式被用来解决形如ax^2 + bx + c = 0的二次方程。
这个公式提供了方程的根或解，也就是使方程等于零的x的值。
理解如何应用一元二次方程求根公式对于在代数学和更高级数学中取得成功至关重要。
The quadratic formula states that the solutions to the quadratic equation ax^2 + bx + c = 0 are given by the formula x = (-b ± √(b^2 - 4ac))/(2a). This formula involves taking the square root of the discriminant, which is b^2 - 4ac, and plugging it into the equation with the coefficients a, b, and c. The ± sign indicates that there are two possible solutions, one with the positive square root and onewith the negative square root. This allows for the calculation of the two roots of the quadratic equation.一元二次方程求根公式表明，一元二次方程ax^2 + bx + c = 0的解由公式x = (-b ± √(b^2 - 4ac))/(2a)给出。
Formula Sheet
Speed, Distance and Time: Distance = speed x time Speed = Time =
������������������������������������������������ ������������������������ ������������������������������������������������ ������������������������������ ������������������������������ ������������������������������������������������ ������������������������������ ������������������������
Percentages: Percent means per hundred. To express one quantity as a percentage of another, first write the first quantity as a fraction of the second and then multiply by 100. Profit = S.P. – C.P. Loss = C.P. – S.P. ������������−������������ Profit percentage = ������������ × 100 Loss percentage =
____________________________________________________________________________________
Mathematics - important points and formulas 2009 Leona :-) mohd.hilmy@ Page 1 of 21 1
高考数学中的非线性方程组解析技巧
高考数学中的非线性方程组解析技巧数学是高考必考的科目，而数学中解析几何的一些内容，如直线、平面、圆锥曲线等知识点会涉及到非线性方程组的解法。
如何解决非线性方程组成为考生必须掌握的考点之一。
非线性方程组的解题需要逐步推导出未知量的值，而其中解析的技巧必不可少。
本篇文章将介绍一些高考数学中的非线性方程组解析技巧。
I. 消元法在高考中，消元法是求解一元或多元非线性方程组的常用方法。
以 $n$ 元非线性方程组为例：$$ \begin{cases} F_1(x_1, x_2, ..., x_n) = 0 \\ F_2(x_1, x_2, ..., x_n) = 0 \\ ... \\ F_n(x_1, x_2, ..., x_n) = 0 \end{cases} $$通过消元法，我们可以将复杂的方程组转化为简单的一元方程。
例如，假设我们要解决如下非线性方程组：$$\begin{cases} x^2+y^2=1 \\ x+y=1 \end{cases} $$We can solve this system of equations by using the elimination method. Adding the equations together, we get:$$x^2 + 2xy + y^2 = 2$$Since $x^2+y^2=1$, we can substitute this into the above equation and obtain:$$2xy = 1$$Then, we can substitute $y=1-x$ into the above equation and obtain:$$2x(1-x) = 1$$This is a quadratic equation that we can solve using the quadratic formula:$$x^2 - x + \frac{1}{2} = 0$$Solving the above quadratic equation, we get:$$x = \frac{1 \pm \sqrt{3}}{2}$$Substituting these values of $x$ into $y=1-x$, we get:$$(x, y) = \left(\frac{1+\sqrt{3}}{2}, \frac{1-\sqrt{3}}{2}\right) \text{ and } \left(\frac{1-\sqrt{3}}{2}, \frac{1+\sqrt{3}}{2}\right)$$消元法可谓是非线性方程组解法的基础，要牢牢掌握。
Completing the square
When a trinomial is a perfect square, there is a relationship between the coefficient of the x-term and the constant term.
X2 +6x+9
x2–8x+16
Divide the coefficient of the x-term by 2, then square the result to get the constant term.
An expression in the form x2+bx is not a perfect square. However, you can use the relationship shown above to add a term to x2+bx to form a trinomial that is a perfect square. This is called completing the square.
Write in the form x2 + bx = c.
Step 2
.
Step 3 x2 – 4x + 4 = 6 + 4
Complete the square.
Step 4 (x – 2)2 = 10
Factor and simplify.
Step 5 x – 2 = ± √10
Take the square root
100 4 25/4
Check It Out!
Complete the square to form a perfect square trinomial.
Solving Equations
3 x 2 2 x 1 (3 x 1)(x 1)
is an identity
An identity is true for all values of the unknown.
( Identities are sometimes written with
instead of = )
( x 2) 2 3 x2 3 x 2 3
N.B. 2 Solutions!
These answers are exact but can be given as approximate decimals.
Solving Equations Solving Quadratic Equations The method used in the last example can be generalised to give us a formula which is easier to use 2 when the coefficient of x is not 1 The formula will be proved but you don’t need to know the proof. However, you must memorise the result.
EXCEPTION: If there is no ‘x’ term write the equation as x 2 c and square root. Try to factorise • Common Factors • Trinomial factors If there are factors, factorise and solve If there are no factors, complete the square ( if a = 1 ) or use the formula
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
x
b 2a
2
b2 4ac 4a2
Add the result to both sides. Simplify Multiply by common denominator Simplify
x
b
2
2a
b2 4ac 4a2
Square root both sides
x b b2 4ac
x = 5 or x = –4
Write as two equations. Solve each equation.
Check It Out! Solve using the Quadratic Formula. –3x2 + 5x + 2 = 0 –3x2 + 5x + 2 = 0 Identify a, b, and c.
Use the quadratic formula.
Substitute 1 for a, –1 for b, and –20 for c.
Simplify.
Example 1B Continued
Solve using the Quadratic Formula. x2 = x + 20
Simplify.
In the previous lesson, you completed the square to solve quadratic equations. If you complete the square of ax2 + bx + c = 0, you can derive the Quadratic Formula. The Quadratic Formula is the only method that can be used to solve any quadratic equation.
x = – or x = 2 Solve each equation.
Check It Out!
Solve using the Quadratic Formula. 2 – 5x2 = –9x
(–5)x2 + 9x + (2) = 0
Write in standard form. Identify a, b, and c
x b b2 4ac 2a
MEMORIZE!!!!
Example 1A: Using the Quadratic Formula Solve using the Quadratic Formula.
6x2 + 5x – 4 = 0 6x2 + 5x + (–4) = 0 Identify a, b, and c.
Use the Quadratic Formula. Substitute –5 for a, 9 for b, and 2 for c. Simplify.
Check It Out!
Solve using the Quadratic Formula. 2 – 5x2 = –9x Simplify.
Solve each equation.
Example 1B: Using the Quadratic Formula
Solve using the Quadratic Formula.
x2 = x + 20 1x2+(–1x)+(–20)=0
Write in standard form. Identify a, b, and c.
2a
2a
Simplify
x b b b 2a 2a 2a
b2 4ac 2a
Common denominator/subtract from both sides
x b b2 4ac 2a
Simplify
Quadratic Formula
●The solutions of a quadratic equation of the form ax2 + bx + c with a≠0 are given by this formula:
x 2
b
x
•
1
2
c
Subtract c/a on both sides.
a 2
a Multiply by ½ and square the result.
x2 b
x
b
2
c
b
2
a 2a a 2a
x
b 2a
2
b2 4a2
c a
x
b 2a
2
b2 4a2
c a
•
4a 4a
THE QUADRATIC FORMULA
b b2 4ac 2a
Objectives
Solve quadratic equations by using the Quadratic Formula. Determine the number of solutions of a quadratic equation by using the discriminant.
Use the Quadratic Formula.
Substitute –3 for a, 5 for b, and 2 for c.
Simplify.
Check It Out! Solve using the Quadratic Formula.
–3x2 + 5x + 2 = 0 Simplify. Write as two equations.
Write as two equations.
x = – or x =2
Solve each equation.
Derive the quadratic formula from ax2+bx+c=0(a≠0)
ax2 bx c 0
General form of a quadratic equation.
ax2 bx c 0 a aaa
Divide all by a
x2 bx c 0 aa
Simplify
Use the Quadratic Formula. Substitute 6 for a, 5 for b, and –4 for c.
Simplify.
Example 1A Continued
Solve using the Quadratic Formula. 6x2 + 5x – 4 = 0 Simplify. Write as two equations.