Solve (20-2x)(14-2x)=160

$$(20-2x)(14-2x)=160$$

Solve for x

$$x=15$$

$$x=2$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 20-2x by 14-2x and combine like terms.

$$280-68x+4x^{2}=160$$

Subtract 160 from both sides.

$$280-68x+4x^{2}-160=0$$

Subtract 160 from 280 to get 120.

$$120-68x+4x^{2}=0$$

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.

$$4x^{2}-68x+120=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -68 for b, and 120 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-68\right)±\sqrt{\left(-68\right)^{2}-4\times 4\times 120}}{2\times 4}$$

Square -68.

$$x=\frac{-\left(-68\right)±\sqrt{4624-4\times 4\times 120}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-68\right)±\sqrt{4624-16\times 120}}{2\times 4}$$

Multiply -16 times 120.

$$x=\frac{-\left(-68\right)±\sqrt{4624-1920}}{2\times 4}$$

Add 4624 to -1920.

$$x=\frac{-\left(-68\right)±\sqrt{2704}}{2\times 4}$$

Take the square root of 2704.

$$x=\frac{-\left(-68\right)±52}{2\times 4}$$

The opposite of -68 is 68.

$$x=\frac{68±52}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{68±52}{8}$$

Now solve the equation x=\frac{68±52}{8} when ± is plus. Add 68 to 52.

$$x=\frac{120}{8}$$

Divide 120 by 8.

$$x=15$$

Now solve the equation x=\frac{68±52}{8} when ± is minus. Subtract 52 from 68.

$$x=\frac{16}{8}$$

Divide 16 by 8.

$$x=2$$

The equation is now solved.

$$x=15 x=2$$

Steps for Completing the Square

Use the distributive property to multiply 20-2x by 14-2x and combine like terms.

$$280-68x+4x^{2}=160$$

Subtract 280 from both sides.

$$-68x+4x^{2}=160-280$$

Subtract 280 from 160 to get -120.

$$-68x+4x^{2}=-120$$

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.

$$4x^{2}-68x=-120$$

Divide both sides by 4.

$$\frac{4x^{2}-68x}{4}=-\frac{120}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\left(-\frac{68}{4}\right)x=-\frac{120}{4}$$

Divide -68 by 4.

$$x^{2}-17x=-\frac{120}{4}$$

Divide -120 by 4.

$$x^{2}-17x=-30$$

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.

$$x^{2}-17x+\left(-\frac{17}{2}\right)^{2}=-30+\left(-\frac{17}{2}\right)^{2}$$

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

$$x^{2}-17x+\frac{289}{4}=-30+\frac{289}{4}$$

Add -30 to \frac{289}{4}.

$$x^{2}-17x+\frac{289}{4}=\frac{169}{4}$$

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}.

$$\left(x-\frac{17}{2}\right)^{2}=\frac{169}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{17}{2}\right)^{2}}=\sqrt{\frac{169}{4}}$$

Simplify.

$$x-\frac{17}{2}=\frac{13}{2} x-\frac{17}{2}=-\frac{13}{2}$$

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

$$x=15 x=2$$