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15-\left(x^{2}-2x\right)=0
Use the distributive property to multiply x by x-2.
15-x^{2}-\left(-2x\right)=0
To find the opposite of x^{2}-2x, find the opposite of each term.
15-x^{2}+2x=0
The opposite of -2x is 2x.
-x^{2}+2x+15=0
Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
a+b=2 ab=-15=-15
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+15. To find a and b, set up a system to be solved.
-1,15 -3,5
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 -15.
-1+15=14 -3+5=2
Calculate the sum for each pair.
a=5 b=-3
The solution is the pair that gives sum 2.
\left(-x^{2}+5x\right)+\left(-3x+15\right)
Rewrite -x^{2}+2x+15 as \left(-x^{2}+5x\right)+\left(-3x+15\right).
-x\left(x-5\right)-3\left(x-5\right)
Factor out -x in the first and -3 in the second group.
\left(x-5\right)\left(-x-3\right)
Factor out common term x-5 by using distributive property.
x=5 x=-3
To find equation solutions, solve x-5=0 and -x-3=0.
15-\left(x^{2}-2x\right)=0
Use the distributive property to multiply x by x-2.
15-x^{2}-\left(-2x\right)=0
To find the opposite of x^{2}-2x, find the opposite of each term.
15-x^{2}+2x=0
The opposite of -2x is 2x.
-x^{2}+2x+15=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{-2±\sqrt{2^{2}-4\left(-1\right)\times 15}}{2\left(-1\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 2 for b, and 15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-2±\sqrt{4-4\left(-1\right)\times 15}}{2\left(-1\right)}
Square 2.
x=\frac{-2±\sqrt{4+4\times 15}}{2\left(-1\right)}
Multiply -4 times -1.
x=\frac{-2±\sqrt{4+60}}{2\left(-1\right)}
Multiply 4 times 15.
x=\frac{-2±\sqrt{64}}{2\left(-1\right)}
Add 4 to 60.
x=\frac{-2±8}{2\left(-1\right)}
Take the square root of 64.
x=\frac{-2±8}{-2}
Multiply 2 times -1.
x=\frac{6}{-2}
Now solve the equation x=\frac{-2±8}{-2} when ± is plus. Add -2 to 8.
x=-3
Divide 6 by -2.
x=-\frac{10}{-2}
Now solve the equation x=\frac{-2±8}{-2} when ± is minus. Subtract 8 from -2.
x=5
Divide -10 by -2.
x=-3 x=5
The equation is now solved.
15-\left(x^{2}-2x\right)=0
Use the distributive property to multiply x by x-2.
15-x^{2}-\left(-2x\right)=0
To find the opposite of x^{2}-2x, find the opposite of each term.
15-x^{2}+2x=0
The opposite of -2x is 2x.
-x^{2}+2x=-15
Subtract 15 from both sides. Anything subtracted from zero gives its negation.
\frac{-x^{2}+2x}{-1}=-\frac{15}{-1}
Divide both sides by -1.
x^{2}+\frac{2}{-1}x=-\frac{15}{-1}
Dividing by -1 undoes the multiplication by -1.
x^{2}-2x=-\frac{15}{-1}
Divide 2 by -1.
x^{2}-2x=15
Divide -15 by -1.
x^{2}-2x+1=15+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=16
Add 15 to 1.
\left(x-1\right)^{2}=16
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{16}
Take the square root of both sides of the equation.
x-1=4 x-1=-4
Simplify.
x=5 x=-3
Add 1 to both sides of the equation.