Solve left(16-2xright)left(9-xright)=112

$$\left( 16-2x \right) \left( 9-x \right) = 112$$

Solve for x

$$x=1$$

$$x=16$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 16-2x by 9-x and combine like terms.

$$144-34x+2x^{2}=112$$

Subtract 112 from both sides.

$$144-34x+2x^{2}-112=0$$

Subtract 112 from 144 to get 32.

$$32-34x+2x^{2}=0$$

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.

$$2x^{2}-34x+32=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -34 for b, and 32 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-34\right)±\sqrt{\left(-34\right)^{2}-4\times 2\times 32}}{2\times 2}$$

Square -34.

$$x=\frac{-\left(-34\right)±\sqrt{1156-4\times 2\times 32}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-34\right)±\sqrt{1156-8\times 32}}{2\times 2}$$

Multiply -8 times 32.

$$x=\frac{-\left(-34\right)±\sqrt{1156-256}}{2\times 2}$$

Add 1156 to -256.

$$x=\frac{-\left(-34\right)±\sqrt{900}}{2\times 2}$$

Take the square root of 900.

$$x=\frac{-\left(-34\right)±30}{2\times 2}$$

The opposite of -34 is 34.

$$x=\frac{34±30}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{34±30}{4}$$

Now solve the equation x=\frac{34±30}{4} when ± is plus. Add 34 to 30.

$$x=\frac{64}{4}$$

Divide 64 by 4.

$$x=16$$

Now solve the equation x=\frac{34±30}{4} when ± is minus. Subtract 30 from 34.

$$x=\frac{4}{4}$$

Divide 4 by 4.

$$x=1$$

The equation is now solved.

$$x=16 x=1$$

Steps for Completing the Square

Use the distributive property to multiply 16-2x by 9-x and combine like terms.

$$144-34x+2x^{2}=112$$

Subtract 144 from both sides.

$$-34x+2x^{2}=112-144$$

Subtract 144 from 112 to get -32.

$$-34x+2x^{2}=-32$$

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.

$$2x^{2}-34x=-32$$

Divide both sides by 2.

$$\frac{2x^{2}-34x}{2}=-\frac{32}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{34}{2}\right)x=-\frac{32}{2}$$

Divide -34 by 2.

$$x^{2}-17x=-\frac{32}{2}$$

Divide -32 by 2.

$$x^{2}-17x=-16$$

Divide -17, the coefficient of the x term, by 2 to get -\frac{17}{2}. Then add the square of -\frac{17}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-17x+\left(-\frac{17}{2}\right)^{2}=-16+\left(-\frac{17}{2}\right)^{2}$$

Square -\frac{17}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-17x+\frac{289}{4}=-16+\frac{289}{4}$$

Add -16 to \frac{289}{4}.

$$x^{2}-17x+\frac{289}{4}=\frac{225}{4}$$

Factor x^{2}-17x+\frac{289}{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}.

$$\left(x-\frac{17}{2}\right)^{2}=\frac{225}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{17}{2}\right)^{2}}=\sqrt{\frac{225}{4}}$$

Simplify.

$$x-\frac{17}{2}=\frac{15}{2} x-\frac{17}{2}=-\frac{15}{2}$$

Add \frac{17}{2} to both sides of the equation.

$$x=16 x=1$$