Solve for x
x=-13
x=4
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a+b=9 ab=-52
To solve the equation, factor x^{2}+9x-52 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.
-1,52 -2,26 -4,13
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 -52.
-1+52=51 -2+26=24 -4+13=9
Calculate the sum for each pair.
a=-4 b=13
The solution is the pair that gives sum 9.
\left(x-4\right)\left(x+13\right)
Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.
x=4 x=-13
To find equation solutions, solve x-4=0 and x+13=0.
a+b=9 ab=1\left(-52\right)=-52
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-52. To find a and b, set up a system to be solved.
-1,52 -2,26 -4,13
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 -52.
-1+52=51 -2+26=24 -4+13=9
Calculate the sum for each pair.
a=-4 b=13
The solution is the pair that gives sum 9.
\left(x^{2}-4x\right)+\left(13x-52\right)
Rewrite x^{2}+9x-52 as \left(x^{2}-4x\right)+\left(13x-52\right).
x\left(x-4\right)+13\left(x-4\right)
Factor out x in the first and 13 in the second group.
\left(x-4\right)\left(x+13\right)
Factor out common term x-4 by using distributive property.
x=4 x=-13
To find equation solutions, solve x-4=0 and x+13=0.
x^{2}+9x-52=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{-9±\sqrt{9^{2}-4\left(-52\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 9 for b, and -52 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-9±\sqrt{81-4\left(-52\right)}}{2}
Square 9.
x=\frac{-9±\sqrt{81+208}}{2}
Multiply -4 times -52.
x=\frac{-9±\sqrt{289}}{2}
Add 81 to 208.
x=\frac{-9±17}{2}
Take the square root of 289.
x=\frac{8}{2}
Now solve the equation x=\frac{-9±17}{2} when ± is plus. Add -9 to 17.
x=4
Divide 8 by 2.
x=-\frac{26}{2}
Now solve the equation x=\frac{-9±17}{2} when ± is minus. Subtract 17 from -9.
x=-13
Divide -26 by 2.
x=4 x=-13
The equation is now solved.
x^{2}+9x-52=0
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.
x^{2}+9x-52-\left(-52\right)=-\left(-52\right)
Add 52 to both sides of the equation.
x^{2}+9x=-\left(-52\right)
Subtracting -52 from itself leaves 0.
x^{2}+9x=52
Subtract -52 from 0.
x^{2}+9x+\left(\frac{9}{2}\right)^{2}=52+\left(\frac{9}{2}\right)^{2}
Divide 9, the coefficient of the x term, by 2 to get \frac{9}{2}. Then add the square of \frac{9}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+9x+\frac{81}{4}=52+\frac{81}{4}
Square \frac{9}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}+9x+\frac{81}{4}=\frac{289}{4}
Add 52 to \frac{81}{4}.
\left(x+\frac{9}{2}\right)^{2}=\frac{289}{4}
Factor x^{2}+9x+\frac{81}{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{9}{2}\right)^{2}}=\sqrt{\frac{289}{4}}
Take the square root of both sides of the equation.
x+\frac{9}{2}=\frac{17}{2} x+\frac{9}{2}=-\frac{17}{2}
Simplify.
x=4 x=-13
Subtract \frac{9}{2} from both sides of the equation.
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