Solve 96*20=(20-x)(126-2x)

$$96 \times 20 = ( 20 - x ) ( 126 - 2 x )$$

Solve for x

$$x = \frac{\sqrt{5689} + 83}{2} \approx 79.212729946$$

$$x = \frac{83 - \sqrt{5689}}{2} \approx 3.787270054$$

Steps Using the Quadratic Formula

Multiply 96 and 20 to get 1920.

$$1920=\left(20-x\right)\left(126-2x\right)$$

Use the distributive property to multiply 20-x by 126-2x and combine like terms.

$$1920=2520-166x+2x^{2}$$

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

$$2520-166x+2x^{2}=1920$$

Subtract 1920 from both sides.

$$2520-166x+2x^{2}-1920=0$$

Subtract 1920 from 2520 to get 600.

$$600-166x+2x^{2}=0$$

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.

$$2x^{2}-166x+600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -166 for b, and 600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-166\right)±\sqrt{\left(-166\right)^{2}-4\times 2\times 600}}{2\times 2}$$

Square -166.

$$x=\frac{-\left(-166\right)±\sqrt{27556-4\times 2\times 600}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-166\right)±\sqrt{27556-8\times 600}}{2\times 2}$$

Multiply -8 times 600.

$$x=\frac{-\left(-166\right)±\sqrt{27556-4800}}{2\times 2}$$

Add 27556 to -4800.

$$x=\frac{-\left(-166\right)±\sqrt{22756}}{2\times 2}$$

Take the square root of 22756.

$$x=\frac{-\left(-166\right)±2\sqrt{5689}}{2\times 2}$$

The opposite of -166 is 166.

$$x=\frac{166±2\sqrt{5689}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{166±2\sqrt{5689}}{4}$$

Now solve the equation x=\frac{166±2\sqrt{5689}}{4} when ± is plus. Add 166 to 2\sqrt{5689}.

$$x=\frac{2\sqrt{5689}+166}{4}$$

Divide 166+2\sqrt{5689} by 4.

$$x=\frac{\sqrt{5689}+83}{2}$$

Now solve the equation x=\frac{166±2\sqrt{5689}}{4} when ± is minus. Subtract 2\sqrt{5689} from 166.

$$x=\frac{166-2\sqrt{5689}}{4}$$

Divide 166-2\sqrt{5689} by 4.

$$x=\frac{83-\sqrt{5689}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{5689}+83}{2} x=\frac{83-\sqrt{5689}}{2}$$

Steps for Completing the Square

Multiply 96 and 20 to get 1920.

$$1920=\left(20-x\right)\left(126-2x\right)$$

Use the distributive property to multiply 20-x by 126-2x and combine like terms.

$$1920=2520-166x+2x^{2}$$

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

$$2520-166x+2x^{2}=1920$$

Subtract 2520 from both sides.

$$-166x+2x^{2}=1920-2520$$

Subtract 2520 from 1920 to get -600.

$$-166x+2x^{2}=-600$$

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.

$$2x^{2}-166x=-600$$

Divide both sides by 2.

$$\frac{2x^{2}-166x}{2}=-\frac{600}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{166}{2}\right)x=-\frac{600}{2}$$

Divide -166 by 2.

$$x^{2}-83x=-\frac{600}{2}$$

Divide -600 by 2.

$$x^{2}-83x=-300$$

Divide -83, the coefficient of the x term, by 2 to get -\frac{83}{2}. Then add the square of -\frac{83}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-83x+\left(-\frac{83}{2}\right)^{2}=-300+\left(-\frac{83}{2}\right)^{2}$$

Square -\frac{83}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-83x+\frac{6889}{4}=-300+\frac{6889}{4}$$

Add -300 to \frac{6889}{4}.

$$x^{2}-83x+\frac{6889}{4}=\frac{5689}{4}$$

Factor x^{2}-83x+\frac{6889}{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}.

$$\left(x-\frac{83}{2}\right)^{2}=\frac{5689}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{83}{2}\right)^{2}}=\sqrt{\frac{5689}{4}}$$

Simplify.

$$x-\frac{83}{2}=\frac{\sqrt{5689}}{2} x-\frac{83}{2}=-\frac{\sqrt{5689}}{2}$$

Add \frac{83}{2} to both sides of the equation.

$$x=\frac{\sqrt{5689}+83}{2} x=\frac{83-\sqrt{5689}}{2}$$