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-x^{2}+13x-36=0
Subtract 36 from both sides.
a+b=13 ab=-\left(-36\right)=36
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-36. To find a and b, set up a system to be solved.
1,36 2,18 3,12 4,9 6,6
Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 36.
1+36=37 2+18=20 3+12=15 4+9=13 6+6=12
Calculate the sum for each pair.
a=9 b=4
The solution is the pair that gives sum 13.
\left(-x^{2}+9x\right)+\left(4x-36\right)
Rewrite -x^{2}+13x-36 as \left(-x^{2}+9x\right)+\left(4x-36\right).
-x\left(x-9\right)+4\left(x-9\right)
Factor out -x in the first and 4 in the second group.
\left(x-9\right)\left(-x+4\right)
Factor out common term x-9 by using distributive property.
x=9 x=4
To find equation solutions, solve x-9=0 and -x+4=0.
-x^{2}+13x=36
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^{2}+13x-36=36-36
Subtract 36 from both sides of the equation.
-x^{2}+13x-36=0
Subtracting 36 from itself leaves 0.
x=\frac{-13±\sqrt{13^{2}-4\left(-1\right)\left(-36\right)}}{2\left(-1\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 13 for b, and -36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-13±\sqrt{169-4\left(-1\right)\left(-36\right)}}{2\left(-1\right)}
Square 13.
x=\frac{-13±\sqrt{169+4\left(-36\right)}}{2\left(-1\right)}
Multiply -4 times -1.
x=\frac{-13±\sqrt{169-144}}{2\left(-1\right)}
Multiply 4 times -36.
x=\frac{-13±\sqrt{25}}{2\left(-1\right)}
Add 169 to -144.
x=\frac{-13±5}{2\left(-1\right)}
Take the square root of 25.
x=\frac{-13±5}{-2}
Multiply 2 times -1.
x=-\frac{8}{-2}
Now solve the equation x=\frac{-13±5}{-2} when ± is plus. Add -13 to 5.
x=4
Divide -8 by -2.
x=-\frac{18}{-2}
Now solve the equation x=\frac{-13±5}{-2} when ± is minus. Subtract 5 from -13.
x=9
Divide -18 by -2.
x=4 x=9
The equation is now solved.
-x^{2}+13x=36
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{-x^{2}+13x}{-1}=\frac{36}{-1}
Divide both sides by -1.
x^{2}+\frac{13}{-1}x=\frac{36}{-1}
Dividing by -1 undoes the multiplication by -1.
x^{2}-13x=\frac{36}{-1}
Divide 13 by -1.
x^{2}-13x=-36
Divide 36 by -1.
x^{2}-13x+\left(-\frac{13}{2}\right)^{2}=-36+\left(-\frac{13}{2}\right)^{2}
Divide -13, the coefficient of the x term, by 2 to get -\frac{13}{2}. Then add the square of -\frac{13}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-13x+\frac{169}{4}=-36+\frac{169}{4}
Square -\frac{13}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}-13x+\frac{169}{4}=\frac{25}{4}
Add -36 to \frac{169}{4}.
\left(x-\frac{13}{2}\right)^{2}=\frac{25}{4}
Factor x^{2}-13x+\frac{169}{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{13}{2}\right)^{2}}=\sqrt{\frac{25}{4}}
Take the square root of both sides of the equation.
x-\frac{13}{2}=\frac{5}{2} x-\frac{13}{2}=-\frac{5}{2}
Simplify.
x=9 x=4
Add \frac{13}{2} to both sides of the equation.