Solve (20-3x)(12-2x)=112

$$(20-3x)(12-2x)=112$$

Solve for x

$$x=2$$

$$x = \frac{32}{3} = 10\frac{2}{3} \approx 10.666666667$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 20-3x by 12-2x and combine like terms.

$$240-76x+6x^{2}=112$$

Subtract 112 from both sides.

$$240-76x+6x^{2}-112=0$$

Subtract 112 from 240 to get 128.

$$128-76x+6x^{2}=0$$

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.

$$6x^{2}-76x+128=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -76 for b, and 128 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-76\right)±\sqrt{\left(-76\right)^{2}-4\times 6\times 128}}{2\times 6}$$

Square -76.

$$x=\frac{-\left(-76\right)±\sqrt{5776-4\times 6\times 128}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-76\right)±\sqrt{5776-24\times 128}}{2\times 6}$$

Multiply -24 times 128.

$$x=\frac{-\left(-76\right)±\sqrt{5776-3072}}{2\times 6}$$

Add 5776 to -3072.

$$x=\frac{-\left(-76\right)±\sqrt{2704}}{2\times 6}$$

Take the square root of 2704.

$$x=\frac{-\left(-76\right)±52}{2\times 6}$$

The opposite of -76 is 76.

$$x=\frac{76±52}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{76±52}{12}$$

Now solve the equation x=\frac{76±52}{12} when ± is plus. Add 76 to 52.

$$x=\frac{128}{12}$$

Reduce the fraction \frac{128}{12} to lowest terms by extracting and canceling out 4.

$$x=\frac{32}{3}$$

Now solve the equation x=\frac{76±52}{12} when ± is minus. Subtract 52 from 76.

$$x=\frac{24}{12}$$

Divide 24 by 12.

$$x=2$$

The equation is now solved.

$$x=\frac{32}{3} x=2$$

Steps for Completing the Square

Use the distributive property to multiply 20-3x by 12-2x and combine like terms.

$$240-76x+6x^{2}=112$$

Subtract 240 from both sides.

$$-76x+6x^{2}=112-240$$

Subtract 240 from 112 to get -128.

$$-76x+6x^{2}=-128$$

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.

$$6x^{2}-76x=-128$$

Divide both sides by 6.

$$\frac{6x^{2}-76x}{6}=-\frac{128}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\left(-\frac{76}{6}\right)x=-\frac{128}{6}$$

Reduce the fraction \frac{-76}{6} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{38}{3}x=-\frac{128}{6}$$

Reduce the fraction \frac{-128}{6} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{38}{3}x=-\frac{64}{3}$$

Divide -\frac{38}{3}, the coefficient of the x term, by 2 to get -\frac{19}{3}. Then add the square of -\frac{19}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{38}{3}x+\left(-\frac{19}{3}\right)^{2}=-\frac{64}{3}+\left(-\frac{19}{3}\right)^{2}$$

Square -\frac{19}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{38}{3}x+\frac{361}{9}=-\frac{64}{3}+\frac{361}{9}$$

Add -\frac{64}{3} to \frac{361}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{38}{3}x+\frac{361}{9}=\frac{169}{9}$$

Factor x^{2}-\frac{38}{3}x+\frac{361}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{19}{3}\right)^{2}=\frac{169}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{19}{3}\right)^{2}}=\sqrt{\frac{169}{9}}$$

Simplify.

$$x-\frac{19}{3}=\frac{13}{3} x-\frac{19}{3}=-\frac{13}{3}$$

Add \frac{19}{3} to both sides of the equation.

$$x=\frac{32}{3} x=2$$