Solve for x
x=-9
x=2
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x^{2}+15x-18-8x=0
Subtract 8x from both sides.
x^{2}+7x-18=0
Combine 15x and -8x to get 7x.
a+b=7 ab=-18
To solve the equation, factor x^{2}+7x-18 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.
-1,18 -2,9 -3,6
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 -18.
-1+18=17 -2+9=7 -3+6=3
Calculate the sum for each pair.
a=-2 b=9
The solution is the pair that gives sum 7.
\left(x-2\right)\left(x+9\right)
Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.
x=2 x=-9
To find equation solutions, solve x-2=0 and x+9=0.
x^{2}+15x-18-8x=0
Subtract 8x from both sides.
x^{2}+7x-18=0
Combine 15x and -8x to get 7x.
a+b=7 ab=1\left(-18\right)=-18
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-18. To find a and b, set up a system to be solved.
-1,18 -2,9 -3,6
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 -18.
-1+18=17 -2+9=7 -3+6=3
Calculate the sum for each pair.
a=-2 b=9
The solution is the pair that gives sum 7.
\left(x^{2}-2x\right)+\left(9x-18\right)
Rewrite x^{2}+7x-18 as \left(x^{2}-2x\right)+\left(9x-18\right).
x\left(x-2\right)+9\left(x-2\right)
Factor out x in the first and 9 in the second group.
\left(x-2\right)\left(x+9\right)
Factor out common term x-2 by using distributive property.
x=2 x=-9
To find equation solutions, solve x-2=0 and x+9=0.
x^{2}+15x-18-8x=0
Subtract 8x from both sides.
x^{2}+7x-18=0
Combine 15x and -8x to get 7x.
x=\frac{-7±\sqrt{7^{2}-4\left(-18\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 7 for b, and -18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-7±\sqrt{49-4\left(-18\right)}}{2}
Square 7.
x=\frac{-7±\sqrt{49+72}}{2}
Multiply -4 times -18.
x=\frac{-7±\sqrt{121}}{2}
Add 49 to 72.
x=\frac{-7±11}{2}
Take the square root of 121.
x=\frac{4}{2}
Now solve the equation x=\frac{-7±11}{2} when ± is plus. Add -7 to 11.
x=2
Divide 4 by 2.
x=-\frac{18}{2}
Now solve the equation x=\frac{-7±11}{2} when ± is minus. Subtract 11 from -7.
x=-9
Divide -18 by 2.
x=2 x=-9
The equation is now solved.
x^{2}+15x-18-8x=0
Subtract 8x from both sides.
x^{2}+7x-18=0
Combine 15x and -8x to get 7x.
x^{2}+7x=18
Add 18 to both sides. Anything plus zero gives itself.
x^{2}+7x+\left(\frac{7}{2}\right)^{2}=18+\left(\frac{7}{2}\right)^{2}
Divide 7, the coefficient of the x term, by 2 to get \frac{7}{2}. Then add the square of \frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+7x+\frac{49}{4}=18+\frac{49}{4}
Square \frac{7}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}+7x+\frac{49}{4}=\frac{121}{4}
Add 18 to \frac{49}{4}.
\left(x+\frac{7}{2}\right)^{2}=\frac{121}{4}
Factor x^{2}+7x+\frac{49}{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+\frac{7}{2}\right)^{2}}=\sqrt{\frac{121}{4}}
Take the square root of both sides of the equation.
x+\frac{7}{2}=\frac{11}{2} x+\frac{7}{2}=-\frac{11}{2}
Simplify.
x=2 x=-9
Subtract \frac{7}{2} from both sides of the equation.
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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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