Solve 32x-x^2=240

$$32 x - x ^ { 2 } = 240$$

Solve for x

$$x=12$$

$$x=20$$

Steps Using Factoring By Grouping

Subtract 240 from both sides.

$$32x-x^{2}-240=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-x^{2}+32x-240=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-240. To find a and b, set up a system to be solved.

$$a+b=32 ab=-\left(-240\right)=240$$

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 240.

$$1,240 2,120 3,80 4,60 5,48 6,40 8,30 10,24 12,20 15,16$$

Calculate the sum for each pair.

$$1+240=241 2+120=122 3+80=83 4+60=64 5+48=53 6+40=46 8+30=38 10+24=34 12+20=32 15+16=31$$

The solution is the pair that gives sum 32.

$$a=20 b=12$$

Rewrite -x^{2}+32x-240 as \left(-x^{2}+20x\right)+\left(12x-240\right).

$$\left(-x^{2}+20x\right)+\left(12x-240\right)$$

Factor out -x in the first and 12 in the second group.

$$-x\left(x-20\right)+12\left(x-20\right)$$

Factor out common term x-20 by using distributive property.

$$\left(x-20\right)\left(-x+12\right)$$

To find equation solutions, solve x-20=0 and -x+12=0.

$$x=20 x=12$$

Steps Using the Quadratic Formula

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.

$$-x^{2}+32x=240$$

Subtract 240 from both sides of the equation.

$$-x^{2}+32x-240=240-240$$

Subtracting 240 from itself leaves 0.

$$-x^{2}+32x-240=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 32 for b, and -240 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-32±\sqrt{32^{2}-4\left(-1\right)\left(-240\right)}}{2\left(-1\right)}$$

Square 32.

$$x=\frac{-32±\sqrt{1024-4\left(-1\right)\left(-240\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-32±\sqrt{1024+4\left(-240\right)}}{2\left(-1\right)}$$

Multiply 4 times -240.

$$x=\frac{-32±\sqrt{1024-960}}{2\left(-1\right)}$$

Add 1024 to -960.

$$x=\frac{-32±\sqrt{64}}{2\left(-1\right)}$$

Take the square root of 64.

$$x=\frac{-32±8}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-32±8}{-2}$$

Now solve the equation x=\frac{-32±8}{-2} when ± is plus. Add -32 to 8.

$$x=-\frac{24}{-2}$$

Divide -24 by -2.

$$x=12$$

Now solve the equation x=\frac{-32±8}{-2} when ± is minus. Subtract 8 from -32.

$$x=-\frac{40}{-2}$$

Divide -40 by -2.

$$x=20$$

The equation is now solved.

$$x=12 x=20$$

Steps for Completing the Square

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.

$$-x^{2}+32x=240$$

Divide both sides by -1.

$$\frac{-x^{2}+32x}{-1}=\frac{240}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{32}{-1}x=\frac{240}{-1}$$

Divide 32 by -1.

$$x^{2}-32x=\frac{240}{-1}$$

Divide 240 by -1.

$$x^{2}-32x=-240$$

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.

$$x^{2}-32x+\left(-16\right)^{2}=-240+\left(-16\right)^{2}$$

Square -16.

$$x^{2}-32x+256=-240+256$$

Add -240 to 256.

$$x^{2}-32x+256=16$$

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}.

$$\left(x-16\right)^{2}=16$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-16\right)^{2}}=\sqrt{16}$$

Simplify.

$$x-16=4 x-16=-4$$

Add 16 to both sides of the equation.

$$x=20 x=12$$