一元方程求根公式
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solve ax+b=0 for x
Isolate terms with x to the left hand side. Solve for x.
solve ax^2+bx+c=0 for x
Write the quadratic equation in standard form.
２
Solve the quadratic equation by completing the square.
Take one half of the coefficient of
x and square it,then add it to both sides.
Factor the left hand side.
Eliminate the exponent on the left hand side.
Look at the first equation:Solve for x.
Look at the second equation:Solve for x.
solve ax^3+bx^2+cx+d=0 for x
Look for a simple substitution that eliminates
the quadratic term of a x3 b x2 c x d.
Write the cubic polynomial on the left hand side in standard
Write the cubic equation in standard form.
Change coordinates by substituting y z Κ
z
,where
Κis a constant value that will be determined later.
Transform the rational equation into a polynomial equation
Find an appropriate value forΚ
in order to make the coefficients of z2and z4both
４
Solve for u.
Perform back substitution on u
1
54a3
9a b c 2b3 27a2d
33a34a c3 b2c2 4b3d 18a b c d 27a2d2
a4
.
Move everything to the left hand side.
Divide by an appropriate factor to make the constant Write 54a3z3 9a b c 2b3 27a2d
33a34a c3 b2c2 4b3d 18a b c d 27a2d2
a4
with a common power.
Simplify 54a3z3 9a b c 2b3 27a2d 33a ６
Simplify 54a3z3 9a b c 2b3 27a2d 33a
4a c3 b2c2 4b3d 18a b c d 27a2d2
1 0by making a substitution.
Solve for v.
Look at the first equation:Perform back substitution
Solve for z.
８
Perform back substitution on
1
9a b c 2b3 27a2d 33a
z
323a
4a c3 b2c2 4b3d 18a b c d 27a2d2^ 1
3 .
Solve for y.
Perform back substitution on
Perform back substitution on
3 b2 3a c 3a9a b c 2b3 27a2d 33a
y 2
4a c3 b2c2 4b3d 18a b c d 27a2d2
1
^ 1 3
323a
9a b c 2b3 27a2d 33a
4a c3 b2c2 4b3d 18a b c d 27a2d2^
1 3 .
Solve for x.
Look at the second equation:
１０
Look at the second equation:
3.
Perform back substitution on v 1
Solve for z.
Perform back substitution on
z
1
3a
1
2
39a b c 2b3 27a2d 33a
4a c3 b2c2 4b3d 18a b c d 27a2d2^ 1 3 .
Solve for y.
Perform back substitution on y 1 2 32
3 b2 3a c
3a9a b c 2b3 27a2d 33a
4a c3 b2c2 4b3d 18a b c d 27a2d2
^ 1 3
1 3a
1 2
39a b c 2b3 27a2d 33a
4a c3 b2c2 4b3d 18a b c d 27a2d2^ 1 3 .