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32x-x^{2}-112-16=103
Use the distributive property to multiply x-4 by 28-x and combine like terms.
32x-x^{2}-128=103
Subtract 16 from -112 to get -128.
32x-x^{2}-128-103=0
Subtract 103 from both sides.
32x-x^{2}-231=0
Subtract 103 from -128 to get -231.
-x^{2}+32x-231=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{-32±\sqrt{32^{2}-4\left(-1\right)\left(-231\right)}}{2\left(-1\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 32 for b, and -231 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-32±\sqrt{1024-4\left(-1\right)\left(-231\right)}}{2\left(-1\right)}
Square 32.
x=\frac{-32±\sqrt{1024+4\left(-231\right)}}{2\left(-1\right)}
Multiply -4 times -1.
x=\frac{-32±\sqrt{1024-924}}{2\left(-1\right)}
Multiply 4 times -231.
x=\frac{-32±\sqrt{100}}{2\left(-1\right)}
Add 1024 to -924.
x=\frac{-32±10}{2\left(-1\right)}
Take the square root of 100.
x=\frac{-32±10}{-2}
Multiply 2 times -1.
x=-\frac{22}{-2}
Now solve the equation x=\frac{-32±10}{-2} when ± is plus. Add -32 to 10.
x=11
Divide -22 by -2.
x=-\frac{42}{-2}
Now solve the equation x=\frac{-32±10}{-2} when ± is minus. Subtract 10 from -32.
x=21
Divide -42 by -2.
x=11 x=21
The equation is now solved.
32x-x^{2}-112-16=103
Use the distributive property to multiply x-4 by 28-x and combine like terms.
32x-x^{2}-128=103
Subtract 16 from -112 to get -128.
32x-x^{2}=103+128
Add 128 to both sides.
32x-x^{2}=231
Add 103 and 128 to get 231.
-x^{2}+32x=231
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}+32x}{-1}=\frac{231}{-1}
Divide both sides by -1.
x^{2}+\frac{32}{-1}x=\frac{231}{-1}
Dividing by -1 undoes the multiplication by -1.
x^{2}-32x=\frac{231}{-1}
Divide 32 by -1.
x^{2}-32x=-231
Divide 231 by -1.
x^{2}-32x+\left(-16\right)^{2}=-231+\left(-16\right)^{2}
Divide -32, the coefficient of the x term, by 2 to get -16. Then add the square of -16 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-32x+256=-231+256
Square -16.
x^{2}-32x+256=25
Add -231 to 256.
\left(x-16\right)^{2}=25
Factor x^{2}-32x+256. 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-16\right)^{2}}=\sqrt{25}
Take the square root of both sides of the equation.
x-16=5 x-16=-5
Simplify.
x=21 x=11
Add 16 to both sides of the equation.