Solve 60=(12-2x)(10-2x)

$$60 = ( 12 - 2 x ) ( 10 - 2 x )$$

Solve for x

$$x = \frac{11 - \sqrt{61}}{2} \approx 1.594875162$$

$$x = \frac{\sqrt{61} + 11}{2} \approx 9.405124838$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 12-2x by 10-2x and combine like terms.

$$60=120-44x+4x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$120-44x+4x^{2}=60$$

Subtract 60 from both sides.

$$120-44x+4x^{2}-60=0$$

Subtract 60 from 120 to get 60.

$$60-44x+4x^{2}=0$$

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.

$$4x^{2}-44x+60=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -44 for b, and 60 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-44\right)±\sqrt{\left(-44\right)^{2}-4\times 4\times 60}}{2\times 4}$$

Square -44.

$$x=\frac{-\left(-44\right)±\sqrt{1936-4\times 4\times 60}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-44\right)±\sqrt{1936-16\times 60}}{2\times 4}$$

Multiply -16 times 60.

$$x=\frac{-\left(-44\right)±\sqrt{1936-960}}{2\times 4}$$

Add 1936 to -960.

$$x=\frac{-\left(-44\right)±\sqrt{976}}{2\times 4}$$

Take the square root of 976.

$$x=\frac{-\left(-44\right)±4\sqrt{61}}{2\times 4}$$

The opposite of -44 is 44.

$$x=\frac{44±4\sqrt{61}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{44±4\sqrt{61}}{8}$$

Now solve the equation x=\frac{44±4\sqrt{61}}{8} when ± is plus. Add 44 to 4\sqrt{61}.

$$x=\frac{4\sqrt{61}+44}{8}$$

Divide 44+4\sqrt{61} by 8.

$$x=\frac{\sqrt{61}+11}{2}$$

Now solve the equation x=\frac{44±4\sqrt{61}}{8} when ± is minus. Subtract 4\sqrt{61} from 44.

$$x=\frac{44-4\sqrt{61}}{8}$$

Divide 44-4\sqrt{61} by 8.

$$x=\frac{11-\sqrt{61}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{61}+11}{2} x=\frac{11-\sqrt{61}}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 12-2x by 10-2x and combine like terms.

$$60=120-44x+4x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$120-44x+4x^{2}=60$$

Subtract 120 from both sides.

$$-44x+4x^{2}=60-120$$

Subtract 120 from 60 to get -60.

$$-44x+4x^{2}=-60$$

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.

$$4x^{2}-44x=-60$$

Divide both sides by 4.

$$\frac{4x^{2}-44x}{4}=-\frac{60}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\left(-\frac{44}{4}\right)x=-\frac{60}{4}$$

Divide -44 by 4.

$$x^{2}-11x=-\frac{60}{4}$$

Divide -60 by 4.

$$x^{2}-11x=-15$$

Divide -11, the coefficient of the x term, by 2 to get -\frac{11}{2}. Then add the square of -\frac{11}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-11x+\left(-\frac{11}{2}\right)^{2}=-15+\left(-\frac{11}{2}\right)^{2}$$

Square -\frac{11}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-11x+\frac{121}{4}=-15+\frac{121}{4}$$

Add -15 to \frac{121}{4}.

$$x^{2}-11x+\frac{121}{4}=\frac{61}{4}$$

Factor x^{2}-11x+\frac{121}{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}.

$$\left(x-\frac{11}{2}\right)^{2}=\frac{61}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{11}{2}\right)^{2}}=\sqrt{\frac{61}{4}}$$

Simplify.

$$x-\frac{11}{2}=\frac{\sqrt{61}}{2} x-\frac{11}{2}=-\frac{\sqrt{61}}{2}$$

Add \frac{11}{2} to both sides of the equation.

$$x=\frac{\sqrt{61}+11}{2} x=\frac{11-\sqrt{61}}{2}$$