Solve for x
x=-8
x=4
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2x^{2}+8-8x=3x^{2}-4x-24
Use the distributive property to multiply 2 by x^{2}+4-4x.
2x^{2}+8-8x-3x^{2}=-4x-24
Subtract 3x^{2} from both sides.
-x^{2}+8-8x=-4x-24
Combine 2x^{2} and -3x^{2} to get -x^{2}.
-x^{2}+8-8x+4x=-24
Add 4x to both sides.
-x^{2}+8-4x=-24
Combine -8x and 4x to get -4x.
-x^{2}+8-4x+24=0
Add 24 to both sides.
-x^{2}+32-4x=0
Add 8 and 24 to get 32.
-x^{2}-4x+32=0
Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
a+b=-4 ab=-32=-32
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+32. To find a and b, set up a system to be solved.
1,-32 2,-16 4,-8
Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -32.
1-32=-31 2-16=-14 4-8=-4
Calculate the sum for each pair.
a=4 b=-8
The solution is the pair that gives sum -4.
\left(-x^{2}+4x\right)+\left(-8x+32\right)
Rewrite -x^{2}-4x+32 as \left(-x^{2}+4x\right)+\left(-8x+32\right).
x\left(-x+4\right)+8\left(-x+4\right)
Factor out x in the first and 8 in the second group.
\left(-x+4\right)\left(x+8\right)
Factor out common term -x+4 by using distributive property.
x=4 x=-8
To find equation solutions, solve -x+4=0 and x+8=0.
2x^{2}+8-8x=3x^{2}-4x-24
Use the distributive property to multiply 2 by x^{2}+4-4x.
2x^{2}+8-8x-3x^{2}=-4x-24
Subtract 3x^{2} from both sides.
-x^{2}+8-8x=-4x-24
Combine 2x^{2} and -3x^{2} to get -x^{2}.
-x^{2}+8-8x+4x=-24
Add 4x to both sides.
-x^{2}+8-4x=-24
Combine -8x and 4x to get -4x.
-x^{2}+8-4x+24=0
Add 24 to both sides.
-x^{2}+32-4x=0
Add 8 and 24 to get 32.
-x^{2}-4x+32=0
All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.
x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-1\right)\times 32}}{2\left(-1\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -4 for b, and 32 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-4\right)±\sqrt{16-4\left(-1\right)\times 32}}{2\left(-1\right)}
Square -4.
x=\frac{-\left(-4\right)±\sqrt{16+4\times 32}}{2\left(-1\right)}
Multiply -4 times -1.
x=\frac{-\left(-4\right)±\sqrt{16+128}}{2\left(-1\right)}
Multiply 4 times 32.
x=\frac{-\left(-4\right)±\sqrt{144}}{2\left(-1\right)}
Add 16 to 128.
x=\frac{-\left(-4\right)±12}{2\left(-1\right)}
Take the square root of 144.
x=\frac{4±12}{2\left(-1\right)}
The opposite of -4 is 4.
x=\frac{4±12}{-2}
Multiply 2 times -1.
x=\frac{16}{-2}
Now solve the equation x=\frac{4±12}{-2} when ± is plus. Add 4 to 12.
x=-8
Divide 16 by -2.
x=-\frac{8}{-2}
Now solve the equation x=\frac{4±12}{-2} when ± is minus. Subtract 12 from 4.
x=4
Divide -8 by -2.
x=-8 x=4
The equation is now solved.
2x^{2}+8-8x=3x^{2}-4x-24
Use the distributive property to multiply 2 by x^{2}+4-4x.
2x^{2}+8-8x-3x^{2}=-4x-24
Subtract 3x^{2} from both sides.
-x^{2}+8-8x=-4x-24
Combine 2x^{2} and -3x^{2} to get -x^{2}.
-x^{2}+8-8x+4x=-24
Add 4x to both sides.
-x^{2}+8-4x=-24
Combine -8x and 4x to get -4x.
-x^{2}-4x=-24-8
Subtract 8 from both sides.
-x^{2}-4x=-32
Subtract 8 from -24 to get -32.
\frac{-x^{2}-4x}{-1}=-\frac{32}{-1}
Divide both sides by -1.
x^{2}+\left(-\frac{4}{-1}\right)x=-\frac{32}{-1}
Dividing by -1 undoes the multiplication by -1.
x^{2}+4x=-\frac{32}{-1}
Divide -4 by -1.
x^{2}+4x=32
Divide -32 by -1.
x^{2}+4x+2^{2}=32+2^{2}
Divide 4, the coefficient of the x term, by 2 to get 2. Then add the square of 2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+4x+4=32+4
Square 2.
x^{2}+4x+4=36
Add 32 to 4.
\left(x+2\right)^{2}=36
Factor x^{2}+4x+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(x+2\right)^{2}}=\sqrt{36}
Take the square root of both sides of the equation.
x+2=6 x+2=-6
Simplify.
x=4 x=-8
Subtract 2 from both sides of the equation.
Examples
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{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
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Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
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