Solve for x
x=4
x=-2
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x^{2}-2x+1-1=8
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.
x^{2}-2x=8
Subtract 1 from 1 to get 0.
x^{2}-2x-8=0
Subtract 8 from both sides.
a+b=-2 ab=-8
To solve the equation, factor x^{2}-2x-8 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.
1,-8 2,-4
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 -8.
1-8=-7 2-4=-2
Calculate the sum for each pair.
a=-4 b=2
The solution is the pair that gives sum -2.
\left(x-4\right)\left(x+2\right)
Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.
x=4 x=-2
To find equation solutions, solve x-4=0 and x+2=0.
x^{2}-2x+1-1=8
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.
x^{2}-2x=8
Subtract 1 from 1 to get 0.
x^{2}-2x-8=0
Subtract 8 from both sides.
a+b=-2 ab=1\left(-8\right)=-8
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-8. To find a and b, set up a system to be solved.
1,-8 2,-4
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 -8.
1-8=-7 2-4=-2
Calculate the sum for each pair.
a=-4 b=2
The solution is the pair that gives sum -2.
\left(x^{2}-4x\right)+\left(2x-8\right)
Rewrite x^{2}-2x-8 as \left(x^{2}-4x\right)+\left(2x-8\right).
x\left(x-4\right)+2\left(x-4\right)
Factor out x in the first and 2 in the second group.
\left(x-4\right)\left(x+2\right)
Factor out common term x-4 by using distributive property.
x=4 x=-2
To find equation solutions, solve x-4=0 and x+2=0.
x^{2}-2x+1-1=8
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.
x^{2}-2x=8
Subtract 1 from 1 to get 0.
x^{2}-2x-8=0
Subtract 8 from both sides.
x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-8\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -2 for b, and -8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-2\right)±\sqrt{4-4\left(-8\right)}}{2}
Square -2.
x=\frac{-\left(-2\right)±\sqrt{4+32}}{2}
Multiply -4 times -8.
x=\frac{-\left(-2\right)±\sqrt{36}}{2}
Add 4 to 32.
x=\frac{-\left(-2\right)±6}{2}
Take the square root of 36.
x=\frac{2±6}{2}
The opposite of -2 is 2.
x=\frac{8}{2}
Now solve the equation x=\frac{2±6}{2} when ± is plus. Add 2 to 6.
x=4
Divide 8 by 2.
x=-\frac{4}{2}
Now solve the equation x=\frac{2±6}{2} when ± is minus. Subtract 6 from 2.
x=-2
Divide -4 by 2.
x=4 x=-2
The equation is now solved.
x^{2}-2x+1-1=8
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.
x^{2}-2x=8
Subtract 1 from 1 to get 0.
x^{2}-2x+1=8+1
Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-2x+1=9
Add 8 to 1.
\left(x-1\right)^{2}=9
Factor x^{2}-2x+1. 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-1\right)^{2}}=\sqrt{9}
Take the square root of both sides of the equation.
x-1=3 x-1=-3
Simplify.
x=4 x=-2
Add 1 to both sides of the equation.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
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
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}