Solve (16-x)(9-x)=112

$$( 16 - x ) ( 9 - x ) = 112$$

Solve for x

$$x = \frac{25 - \sqrt{497}}{2} \approx 1.353251595$$

$$x = \frac{\sqrt{497} + 25}{2} \approx 23.646748405$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 16-x by 9-x and combine like terms.

$$144-25x+x^{2}=112$$

Subtract 112 from both sides.

$$144-25x+x^{2}-112=0$$

Subtract 112 from 144 to get 32.

$$32-25x+x^{2}=0$$

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.

$$x^{2}-25x+32=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -25 for b, and 32 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-25\right)±\sqrt{\left(-25\right)^{2}-4\times 32}}{2}$$

Square -25.

$$x=\frac{-\left(-25\right)±\sqrt{625-4\times 32}}{2}$$

Multiply -4 times 32.

$$x=\frac{-\left(-25\right)±\sqrt{625-128}}{2}$$

Add 625 to -128.

$$x=\frac{-\left(-25\right)±\sqrt{497}}{2}$$

The opposite of -25 is 25.

$$x=\frac{25±\sqrt{497}}{2}$$

Now solve the equation x=\frac{25±\sqrt{497}}{2} when ± is plus. Add 25 to \sqrt{497}.

$$x=\frac{\sqrt{497}+25}{2}$$

Now solve the equation x=\frac{25±\sqrt{497}}{2} when ± is minus. Subtract \sqrt{497} from 25.

$$x=\frac{25-\sqrt{497}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{497}+25}{2} x=\frac{25-\sqrt{497}}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 16-x by 9-x and combine like terms.

$$144-25x+x^{2}=112$$

Subtract 144 from both sides.

$$-25x+x^{2}=112-144$$

Subtract 144 from 112 to get -32.

$$-25x+x^{2}=-32$$

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.

$$x^{2}-25x=-32$$

Divide -25, the coefficient of the x term, by 2 to get -\frac{25}{2}. Then add the square of -\frac{25}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-25x+\left(-\frac{25}{2}\right)^{2}=-32+\left(-\frac{25}{2}\right)^{2}$$

Square -\frac{25}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-25x+\frac{625}{4}=-32+\frac{625}{4}$$

Add -32 to \frac{625}{4}.

$$x^{2}-25x+\frac{625}{4}=\frac{497}{4}$$

Factor x^{2}-25x+\frac{625}{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}.

$$\left(x-\frac{25}{2}\right)^{2}=\frac{497}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{25}{2}\right)^{2}}=\sqrt{\frac{497}{4}}$$

Simplify.

$$x-\frac{25}{2}=\frac{\sqrt{497}}{2} x-\frac{25}{2}=-\frac{\sqrt{497}}{2}$$

Add \frac{25}{2} to both sides of the equation.

$$x=\frac{\sqrt{497}+25}{2} x=\frac{25-\sqrt{497}}{2}$$