Solve 75*20=(75-x)*(110-x)

$$75 \times 20 = ( 75 - x ) \times ( 110 - x )$$

Solve for x

$$x=135$$

$$x=50$$

Steps Using the Quadratic Formula

Multiply 75 and 20 to get 1500.

$$1500=\left(75-x\right)\left(110-x\right)$$

Use the distributive property to multiply 75-x by 110-x and combine like terms.

$$1500=8250-185x+x^{2}$$

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

$$8250-185x+x^{2}=1500$$

Subtract 1500 from both sides.

$$8250-185x+x^{2}-1500=0$$

Subtract 1500 from 8250 to get 6750.

$$6750-185x+x^{2}=0$$

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.

$$x^{2}-185x+6750=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -185 for b, and 6750 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-185\right)±\sqrt{\left(-185\right)^{2}-4\times 6750}}{2}$$

Square -185.

$$x=\frac{-\left(-185\right)±\sqrt{34225-4\times 6750}}{2}$$

Multiply -4 times 6750.

$$x=\frac{-\left(-185\right)±\sqrt{34225-27000}}{2}$$

Add 34225 to -27000.

$$x=\frac{-\left(-185\right)±\sqrt{7225}}{2}$$

Take the square root of 7225.

$$x=\frac{-\left(-185\right)±85}{2}$$

The opposite of -185 is 185.

$$x=\frac{185±85}{2}$$

Now solve the equation x=\frac{185±85}{2} when ± is plus. Add 185 to 85.

$$x=\frac{270}{2}$$

Divide 270 by 2.

$$x=135$$

Now solve the equation x=\frac{185±85}{2} when ± is minus. Subtract 85 from 185.

$$x=\frac{100}{2}$$

Divide 100 by 2.

$$x=50$$

The equation is now solved.

$$x=135 x=50$$

Steps for Completing the Square

Multiply 75 and 20 to get 1500.

$$1500=\left(75-x\right)\left(110-x\right)$$

Use the distributive property to multiply 75-x by 110-x and combine like terms.

$$1500=8250-185x+x^{2}$$

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

$$8250-185x+x^{2}=1500$$

Subtract 8250 from both sides.

$$-185x+x^{2}=1500-8250$$

Subtract 8250 from 1500 to get -6750.

$$-185x+x^{2}=-6750$$

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.

$$x^{2}-185x=-6750$$

Divide -185, the coefficient of the x term, by 2 to get -\frac{185}{2}. Then add the square of -\frac{185}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-185x+\left(-\frac{185}{2}\right)^{2}=-6750+\left(-\frac{185}{2}\right)^{2}$$

Square -\frac{185}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-185x+\frac{34225}{4}=-6750+\frac{34225}{4}$$

Add -6750 to \frac{34225}{4}.

$$x^{2}-185x+\frac{34225}{4}=\frac{7225}{4}$$

Factor x^{2}-185x+\frac{34225}{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}.

$$\left(x-\frac{185}{2}\right)^{2}=\frac{7225}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{185}{2}\right)^{2}}=\sqrt{\frac{7225}{4}}$$

Simplify.

$$x-\frac{185}{2}=\frac{85}{2} x-\frac{185}{2}=-\frac{85}{2}$$

Add \frac{185}{2} to both sides of the equation.

$$x=135 x=50$$