Solve for x
x=-4
x = \frac{8}{5} = 1\frac{3}{5} = 1.6
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\left(x+7\right)\left(x-2\right)-\left(x-7\right)x=18-5x^{2}
Variable x cannot be equal to any of the values -7,7 since division by zero is not defined. Multiply both sides of the equation by \left(x-7\right)\left(x+7\right), the least common multiple of x-7,x+7,x^{2}-49.
x^{2}+5x-14-\left(x-7\right)x=18-5x^{2}
Use the distributive property to multiply x+7 by x-2 and combine like terms.
x^{2}+5x-14-\left(x^{2}-7x\right)=18-5x^{2}
Use the distributive property to multiply x-7 by x.
x^{2}+5x-14-x^{2}+7x=18-5x^{2}
To find the opposite of x^{2}-7x, find the opposite of each term.
5x-14+7x=18-5x^{2}
Combine x^{2} and -x^{2} to get 0.
12x-14=18-5x^{2}
Combine 5x and 7x to get 12x.
12x-14-18=-5x^{2}
Subtract 18 from both sides.
12x-32=-5x^{2}
Subtract 18 from -14 to get -32.
12x-32+5x^{2}=0
Add 5x^{2} to both sides.
5x^{2}+12x-32=0
Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
a+b=12 ab=5\left(-32\right)=-160
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 5x^{2}+ax+bx-32. To find a and b, set up a system to be solved.
-1,160 -2,80 -4,40 -5,32 -8,20 -10,16
Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -160.
-1+160=159 -2+80=78 -4+40=36 -5+32=27 -8+20=12 -10+16=6
Calculate the sum for each pair.
a=-8 b=20
The solution is the pair that gives sum 12.
\left(5x^{2}-8x\right)+\left(20x-32\right)
Rewrite 5x^{2}+12x-32 as \left(5x^{2}-8x\right)+\left(20x-32\right).
x\left(5x-8\right)+4\left(5x-8\right)
Factor out x in the first and 4 in the second group.
\left(5x-8\right)\left(x+4\right)
Factor out common term 5x-8 by using distributive property.
x=\frac{8}{5} x=-4
To find equation solutions, solve 5x-8=0 and x+4=0.
\left(x+7\right)\left(x-2\right)-\left(x-7\right)x=18-5x^{2}
Variable x cannot be equal to any of the values -7,7 since division by zero is not defined. Multiply both sides of the equation by \left(x-7\right)\left(x+7\right), the least common multiple of x-7,x+7,x^{2}-49.
x^{2}+5x-14-\left(x-7\right)x=18-5x^{2}
Use the distributive property to multiply x+7 by x-2 and combine like terms.
x^{2}+5x-14-\left(x^{2}-7x\right)=18-5x^{2}
Use the distributive property to multiply x-7 by x.
x^{2}+5x-14-x^{2}+7x=18-5x^{2}
To find the opposite of x^{2}-7x, find the opposite of each term.
5x-14+7x=18-5x^{2}
Combine x^{2} and -x^{2} to get 0.
12x-14=18-5x^{2}
Combine 5x and 7x to get 12x.
12x-14-18=-5x^{2}
Subtract 18 from both sides.
12x-32=-5x^{2}
Subtract 18 from -14 to get -32.
12x-32+5x^{2}=0
Add 5x^{2} to both sides.
5x^{2}+12x-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{-12±\sqrt{12^{2}-4\times 5\left(-32\right)}}{2\times 5}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 12 for b, and -32 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-12±\sqrt{144-4\times 5\left(-32\right)}}{2\times 5}
Square 12.
x=\frac{-12±\sqrt{144-20\left(-32\right)}}{2\times 5}
Multiply -4 times 5.
x=\frac{-12±\sqrt{144+640}}{2\times 5}
Multiply -20 times -32.
x=\frac{-12±\sqrt{784}}{2\times 5}
Add 144 to 640.
x=\frac{-12±28}{2\times 5}
Take the square root of 784.
x=\frac{-12±28}{10}
Multiply 2 times 5.
x=\frac{16}{10}
Now solve the equation x=\frac{-12±28}{10} when ± is plus. Add -12 to 28.
x=\frac{8}{5}
Reduce the fraction \frac{16}{10} to lowest terms by extracting and canceling out 2.
x=-\frac{40}{10}
Now solve the equation x=\frac{-12±28}{10} when ± is minus. Subtract 28 from -12.
x=-4
Divide -40 by 10.
x=\frac{8}{5} x=-4
The equation is now solved.
\left(x+7\right)\left(x-2\right)-\left(x-7\right)x=18-5x^{2}
Variable x cannot be equal to any of the values -7,7 since division by zero is not defined. Multiply both sides of the equation by \left(x-7\right)\left(x+7\right), the least common multiple of x-7,x+7,x^{2}-49.
x^{2}+5x-14-\left(x-7\right)x=18-5x^{2}
Use the distributive property to multiply x+7 by x-2 and combine like terms.
x^{2}+5x-14-\left(x^{2}-7x\right)=18-5x^{2}
Use the distributive property to multiply x-7 by x.
x^{2}+5x-14-x^{2}+7x=18-5x^{2}
To find the opposite of x^{2}-7x, find the opposite of each term.
5x-14+7x=18-5x^{2}
Combine x^{2} and -x^{2} to get 0.
12x-14=18-5x^{2}
Combine 5x and 7x to get 12x.
12x-14+5x^{2}=18
Add 5x^{2} to both sides.
12x+5x^{2}=18+14
Add 14 to both sides.
12x+5x^{2}=32
Add 18 and 14 to get 32.
5x^{2}+12x=32
Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.
\frac{5x^{2}+12x}{5}=\frac{32}{5}
Divide both sides by 5.
x^{2}+\frac{12}{5}x=\frac{32}{5}
Dividing by 5 undoes the multiplication by 5.
x^{2}+\frac{12}{5}x+\left(\frac{6}{5}\right)^{2}=\frac{32}{5}+\left(\frac{6}{5}\right)^{2}
Divide \frac{12}{5}, the coefficient of the x term, by 2 to get \frac{6}{5}. Then add the square of \frac{6}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{12}{5}x+\frac{36}{25}=\frac{32}{5}+\frac{36}{25}
Square \frac{6}{5} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{12}{5}x+\frac{36}{25}=\frac{196}{25}
Add \frac{32}{5} to \frac{36}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{6}{5}\right)^{2}=\frac{196}{25}
Factor x^{2}+\frac{12}{5}x+\frac{36}{25}. 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+\frac{6}{5}\right)^{2}}=\sqrt{\frac{196}{25}}
Take the square root of both sides of the equation.
x+\frac{6}{5}=\frac{14}{5} x+\frac{6}{5}=-\frac{14}{5}
Simplify.
x=\frac{8}{5} x=-4
Subtract \frac{6}{5} from 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}