Solve (100-x)(50-x)=500*88*32%

$$( 100 - x ) ( 50 - x ) = 500 \times 88 \cdot 32 \%$$

Solve for x

$$x=75-\sqrt{14705}\approx -46.264174429$$

$$x=\sqrt{14705}+75\approx 196.264174429$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 100-x by 50-x and combine like terms.

$$5000-150x+x^{2}=500\times 88\times \frac{32}{100}$$

Multiply 500 and 88 to get 44000.

$$5000-150x+x^{2}=44000\times \frac{32}{100}$$

Reduce the fraction \frac{32}{100} to lowest terms by extracting and canceling out 4.

$$5000-150x+x^{2}=44000\times \frac{8}{25}$$

Multiply 44000 and \frac{8}{25} to get 14080.

$$5000-150x+x^{2}=14080$$

Subtract 14080 from both sides.

$$5000-150x+x^{2}-14080=0$$

Subtract 14080 from 5000 to get -9080.

$$-9080-150x+x^{2}=0$$

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.

$$x^{2}-150x-9080=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -150 for b, and -9080 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-150\right)±\sqrt{\left(-150\right)^{2}-4\left(-9080\right)}}{2}$$

Square -150.

$$x=\frac{-\left(-150\right)±\sqrt{22500-4\left(-9080\right)}}{2}$$

Multiply -4 times -9080.

$$x=\frac{-\left(-150\right)±\sqrt{22500+36320}}{2}$$

Add 22500 to 36320.

$$x=\frac{-\left(-150\right)±\sqrt{58820}}{2}$$

Take the square root of 58820.

$$x=\frac{-\left(-150\right)±2\sqrt{14705}}{2}$$

The opposite of -150 is 150.

$$x=\frac{150±2\sqrt{14705}}{2}$$

Now solve the equation x=\frac{150±2\sqrt{14705}}{2} when ± is plus. Add 150 to 2\sqrt{14705}.

$$x=\frac{2\sqrt{14705}+150}{2}$$

Divide 150+2\sqrt{14705} by 2.

$$x=\sqrt{14705}+75$$

Now solve the equation x=\frac{150±2\sqrt{14705}}{2} when ± is minus. Subtract 2\sqrt{14705} from 150.

$$x=\frac{150-2\sqrt{14705}}{2}$$

Divide 150-2\sqrt{14705} by 2.

$$x=75-\sqrt{14705}$$

The equation is now solved.

$$x=\sqrt{14705}+75 x=75-\sqrt{14705}$$

Steps for Completing the Square

Use the distributive property to multiply 100-x by 50-x and combine like terms.

$$5000-150x+x^{2}=500\times 88\times \frac{32}{100}$$

Multiply 500 and 88 to get 44000.

$$5000-150x+x^{2}=44000\times \frac{32}{100}$$

Reduce the fraction \frac{32}{100} to lowest terms by extracting and canceling out 4.

$$5000-150x+x^{2}=44000\times \frac{8}{25}$$

Multiply 44000 and \frac{8}{25} to get 14080.

$$5000-150x+x^{2}=14080$$

Subtract 5000 from both sides.

$$-150x+x^{2}=14080-5000$$

Subtract 5000 from 14080 to get 9080.

$$-150x+x^{2}=9080$$

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.

$$x^{2}-150x=9080$$

Divide -150, the coefficient of the x term, by 2 to get -75. Then add the square of -75 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-150x+\left(-75\right)^{2}=9080+\left(-75\right)^{2}$$

Square -75.

$$x^{2}-150x+5625=9080+5625$$

Add 9080 to 5625.

$$x^{2}-150x+5625=14705$$

Factor x^{2}-150x+5625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-75\right)^{2}=14705$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-75\right)^{2}}=\sqrt{14705}$$

Simplify.

$$x-75=\sqrt{14705} x-75=-\sqrt{14705}$$

Add 75 to both sides of the equation.

$$x=\sqrt{14705}+75 x=75-\sqrt{14705}$$