Solve (100-x)(50-x)=4416

$$(100-x)(50-x)=4416$$

Solve for x

$$x=146$$

$$x=4$$

Steps Using the Quadratic Formula

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

$$5000-150x+x^{2}=4416$$

Subtract 4416 from both sides.

$$5000-150x+x^{2}-4416=0$$

Subtract 4416 from 5000 to get 584.

$$584-150x+x^{2}=0$$

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.

$$x^{2}-150x+584=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -150 for b, and 584 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-150\right)±\sqrt{\left(-150\right)^{2}-4\times 584}}{2}$$

Square -150.

$$x=\frac{-\left(-150\right)±\sqrt{22500-4\times 584}}{2}$$

Multiply -4 times 584.

$$x=\frac{-\left(-150\right)±\sqrt{22500-2336}}{2}$$

Add 22500 to -2336.

$$x=\frac{-\left(-150\right)±\sqrt{20164}}{2}$$

Take the square root of 20164.

$$x=\frac{-\left(-150\right)±142}{2}$$

The opposite of -150 is 150.

$$x=\frac{150±142}{2}$$

Now solve the equation x=\frac{150±142}{2} when ± is plus. Add 150 to 142.

$$x=\frac{292}{2}$$

Divide 292 by 2.

$$x=146$$

Now solve the equation x=\frac{150±142}{2} when ± is minus. Subtract 142 from 150.

$$x=\frac{8}{2}$$

Divide 8 by 2.

$$x=4$$

The equation is now solved.

$$x=146 x=4$$

Steps for Completing the Square

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

$$5000-150x+x^{2}=4416$$

Subtract 5000 from both sides.

$$-150x+x^{2}=4416-5000$$

Subtract 5000 from 4416 to get -584.

$$-150x+x^{2}=-584$$

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.

$$x^{2}-150x=-584$$

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.

$$x^{2}-150x+\left(-75\right)^{2}=-584+\left(-75\right)^{2}$$

Square -75.

$$x^{2}-150x+5625=-584+5625$$

Add -584 to 5625.

$$x^{2}-150x+5625=5041$$

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}.

$$\left(x-75\right)^{2}=5041$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-75\right)^{2}}=\sqrt{5041}$$

Simplify.

$$x-75=71 x-75=-71$$

Add 75 to both sides of the equation.

$$x=146 x=4$$