Solve x32-x/2=120

$$x \frac{ 32-x }{ 2 } = 120$$

Solve for x

$$x=12$$

$$x=20$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$x\left(32-x\right)=240$$

Use the distributive property to multiply x by 32-x.

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

Subtract 240 from both sides.

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

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.

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

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

Multiply both sides of the equation by 2.

$$x\left(32-x\right)=240$$

Use the distributive property to multiply x by 32-x.

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

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.

$$x=20 x=12$$