Solve for x
x=-4
x=6
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\left(x+3\right)\left(\left(x-3\right)^{2}-\left(x+1\right)^{2}\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}
Variable x cannot be equal to any of the values -3,1,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x-1\right)\left(x+3\right), the least common multiple of x^{2}-4x+3,x^{2}-9,x^{3}-x^{2}-9x+9.
\left(x+3\right)\left(x^{2}-6x+9-\left(x+1\right)^{2}\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.
\left(x+3\right)\left(x^{2}-6x+9-\left(x^{2}+2x+1\right)\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.
\left(x+3\right)\left(x^{2}-6x+9-x^{2}-2x-1\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}
To find the opposite of x^{2}+2x+1, find the opposite of each term.
\left(x+3\right)\left(-6x+9-2x-1\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}
Combine x^{2} and -x^{2} to get 0.
\left(x+3\right)\left(-8x+9-1\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}
Combine -6x and -2x to get -8x.
\left(x+3\right)\left(-8x+8\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}
Subtract 1 from 9 to get 8.
-8x^{2}-16x+24-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}
Use the distributive property to multiply x+3 by -8x+8 and combine like terms.
-8x^{2}-16x+24-\left(x^{2}+3x-4\right)=-17x+52-10x^{2}
Use the distributive property to multiply x-1 by x+4 and combine like terms.
-8x^{2}-16x+24-x^{2}-3x+4=-17x+52-10x^{2}
To find the opposite of x^{2}+3x-4, find the opposite of each term.
-9x^{2}-16x+24-3x+4=-17x+52-10x^{2}
Combine -8x^{2} and -x^{2} to get -9x^{2}.
-9x^{2}-19x+24+4=-17x+52-10x^{2}
Combine -16x and -3x to get -19x.
-9x^{2}-19x+28=-17x+52-10x^{2}
Add 24 and 4 to get 28.
-9x^{2}-19x+28+17x=52-10x^{2}
Add 17x to both sides.
-9x^{2}-2x+28=52-10x^{2}
Combine -19x and 17x to get -2x.
-9x^{2}-2x+28-52=-10x^{2}
Subtract 52 from both sides.
-9x^{2}-2x-24=-10x^{2}
Subtract 52 from 28 to get -24.
-9x^{2}-2x-24+10x^{2}=0
Add 10x^{2} to both sides.
x^{2}-2x-24=0
Combine -9x^{2} and 10x^{2} to get x^{2}.
a+b=-2 ab=-24
To solve the equation, factor x^{2}-2x-24 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,-24 2,-12 3,-8 4,-6
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 -24.
1-24=-23 2-12=-10 3-8=-5 4-6=-2
Calculate the sum for each pair.
a=-6 b=4
The solution is the pair that gives sum -2.
\left(x-6\right)\left(x+4\right)
Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.
x=6 x=-4
To find equation solutions, solve x-6=0 and x+4=0.
\left(x+3\right)\left(\left(x-3\right)^{2}-\left(x+1\right)^{2}\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}
Variable x cannot be equal to any of the values -3,1,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x-1\right)\left(x+3\right), the least common multiple of x^{2}-4x+3,x^{2}-9,x^{3}-x^{2}-9x+9.
\left(x+3\right)\left(x^{2}-6x+9-\left(x+1\right)^{2}\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.
\left(x+3\right)\left(x^{2}-6x+9-\left(x^{2}+2x+1\right)\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.
\left(x+3\right)\left(x^{2}-6x+9-x^{2}-2x-1\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}
To find the opposite of x^{2}+2x+1, find the opposite of each term.
\left(x+3\right)\left(-6x+9-2x-1\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}
Combine x^{2} and -x^{2} to get 0.
\left(x+3\right)\left(-8x+9-1\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}
Combine -6x and -2x to get -8x.
\left(x+3\right)\left(-8x+8\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}
Subtract 1 from 9 to get 8.
-8x^{2}-16x+24-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}
Use the distributive property to multiply x+3 by -8x+8 and combine like terms.
-8x^{2}-16x+24-\left(x^{2}+3x-4\right)=-17x+52-10x^{2}
Use the distributive property to multiply x-1 by x+4 and combine like terms.
-8x^{2}-16x+24-x^{2}-3x+4=-17x+52-10x^{2}
To find the opposite of x^{2}+3x-4, find the opposite of each term.
-9x^{2}-16x+24-3x+4=-17x+52-10x^{2}
Combine -8x^{2} and -x^{2} to get -9x^{2}.
-9x^{2}-19x+24+4=-17x+52-10x^{2}
Combine -16x and -3x to get -19x.
-9x^{2}-19x+28=-17x+52-10x^{2}
Add 24 and 4 to get 28.
-9x^{2}-19x+28+17x=52-10x^{2}
Add 17x to both sides.
-9x^{2}-2x+28=52-10x^{2}
Combine -19x and 17x to get -2x.
-9x^{2}-2x+28-52=-10x^{2}
Subtract 52 from both sides.
-9x^{2}-2x-24=-10x^{2}
Subtract 52 from 28 to get -24.
-9x^{2}-2x-24+10x^{2}=0
Add 10x^{2} to both sides.
x^{2}-2x-24=0
Combine -9x^{2} and 10x^{2} to get x^{2}.
a+b=-2 ab=1\left(-24\right)=-24
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-24. To find a and b, set up a system to be solved.
1,-24 2,-12 3,-8 4,-6
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 -24.
1-24=-23 2-12=-10 3-8=-5 4-6=-2
Calculate the sum for each pair.
a=-6 b=4
The solution is the pair that gives sum -2.
\left(x^{2}-6x\right)+\left(4x-24\right)
Rewrite x^{2}-2x-24 as \left(x^{2}-6x\right)+\left(4x-24\right).
x\left(x-6\right)+4\left(x-6\right)
Factor out x in the first and 4 in the second group.
\left(x-6\right)\left(x+4\right)
Factor out common term x-6 by using distributive property.
x=6 x=-4
To find equation solutions, solve x-6=0 and x+4=0.
\left(x+3\right)\left(\left(x-3\right)^{2}-\left(x+1\right)^{2}\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}
Variable x cannot be equal to any of the values -3,1,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x-1\right)\left(x+3\right), the least common multiple of x^{2}-4x+3,x^{2}-9,x^{3}-x^{2}-9x+9.
\left(x+3\right)\left(x^{2}-6x+9-\left(x+1\right)^{2}\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.
\left(x+3\right)\left(x^{2}-6x+9-\left(x^{2}+2x+1\right)\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.
\left(x+3\right)\left(x^{2}-6x+9-x^{2}-2x-1\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}
To find the opposite of x^{2}+2x+1, find the opposite of each term.
\left(x+3\right)\left(-6x+9-2x-1\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}
Combine x^{2} and -x^{2} to get 0.
\left(x+3\right)\left(-8x+9-1\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}
Combine -6x and -2x to get -8x.
\left(x+3\right)\left(-8x+8\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}
Subtract 1 from 9 to get 8.
-8x^{2}-16x+24-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}
Use the distributive property to multiply x+3 by -8x+8 and combine like terms.
-8x^{2}-16x+24-\left(x^{2}+3x-4\right)=-17x+52-10x^{2}
Use the distributive property to multiply x-1 by x+4 and combine like terms.
-8x^{2}-16x+24-x^{2}-3x+4=-17x+52-10x^{2}
To find the opposite of x^{2}+3x-4, find the opposite of each term.
-9x^{2}-16x+24-3x+4=-17x+52-10x^{2}
Combine -8x^{2} and -x^{2} to get -9x^{2}.
-9x^{2}-19x+24+4=-17x+52-10x^{2}
Combine -16x and -3x to get -19x.
-9x^{2}-19x+28=-17x+52-10x^{2}
Add 24 and 4 to get 28.
-9x^{2}-19x+28+17x=52-10x^{2}
Add 17x to both sides.
-9x^{2}-2x+28=52-10x^{2}
Combine -19x and 17x to get -2x.
-9x^{2}-2x+28-52=-10x^{2}
Subtract 52 from both sides.
-9x^{2}-2x-24=-10x^{2}
Subtract 52 from 28 to get -24.
-9x^{2}-2x-24+10x^{2}=0
Add 10x^{2} to both sides.
x^{2}-2x-24=0
Combine -9x^{2} and 10x^{2} to get x^{2}.
x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-24\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -2 for b, and -24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-2\right)±\sqrt{4-4\left(-24\right)}}{2}
Square -2.
x=\frac{-\left(-2\right)±\sqrt{4+96}}{2}
Multiply -4 times -24.
x=\frac{-\left(-2\right)±\sqrt{100}}{2}
Add 4 to 96.
x=\frac{-\left(-2\right)±10}{2}
Take the square root of 100.
x=\frac{2±10}{2}
The opposite of -2 is 2.
x=\frac{12}{2}
Now solve the equation x=\frac{2±10}{2} when ± is plus. Add 2 to 10.
x=6
Divide 12 by 2.
x=-\frac{8}{2}
Now solve the equation x=\frac{2±10}{2} when ± is minus. Subtract 10 from 2.
x=-4
Divide -8 by 2.
x=6 x=-4
The equation is now solved.
\left(x+3\right)\left(\left(x-3\right)^{2}-\left(x+1\right)^{2}\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}
Variable x cannot be equal to any of the values -3,1,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x-1\right)\left(x+3\right), the least common multiple of x^{2}-4x+3,x^{2}-9,x^{3}-x^{2}-9x+9.
\left(x+3\right)\left(x^{2}-6x+9-\left(x+1\right)^{2}\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.
\left(x+3\right)\left(x^{2}-6x+9-\left(x^{2}+2x+1\right)\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.
\left(x+3\right)\left(x^{2}-6x+9-x^{2}-2x-1\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}
To find the opposite of x^{2}+2x+1, find the opposite of each term.
\left(x+3\right)\left(-6x+9-2x-1\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}
Combine x^{2} and -x^{2} to get 0.
\left(x+3\right)\left(-8x+9-1\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}
Combine -6x and -2x to get -8x.
\left(x+3\right)\left(-8x+8\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}
Subtract 1 from 9 to get 8.
-8x^{2}-16x+24-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}
Use the distributive property to multiply x+3 by -8x+8 and combine like terms.
-8x^{2}-16x+24-\left(x^{2}+3x-4\right)=-17x+52-10x^{2}
Use the distributive property to multiply x-1 by x+4 and combine like terms.
-8x^{2}-16x+24-x^{2}-3x+4=-17x+52-10x^{2}
To find the opposite of x^{2}+3x-4, find the opposite of each term.
-9x^{2}-16x+24-3x+4=-17x+52-10x^{2}
Combine -8x^{2} and -x^{2} to get -9x^{2}.
-9x^{2}-19x+24+4=-17x+52-10x^{2}
Combine -16x and -3x to get -19x.
-9x^{2}-19x+28=-17x+52-10x^{2}
Add 24 and 4 to get 28.
-9x^{2}-19x+28+17x=52-10x^{2}
Add 17x to both sides.
-9x^{2}-2x+28=52-10x^{2}
Combine -19x and 17x to get -2x.
-9x^{2}-2x+28+10x^{2}=52
Add 10x^{2} to both sides.
x^{2}-2x+28=52
Combine -9x^{2} and 10x^{2} to get x^{2}.
x^{2}-2x=52-28
Subtract 28 from both sides.
x^{2}-2x=24
Subtract 28 from 52 to get 24.
x^{2}-2x+1=24+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=25
Add 24 to 1.
\left(x-1\right)^{2}=25
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{25}
Take the square root of both sides of the equation.
x-1=5 x-1=-5
Simplify.
x=6 x=-4
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}