Solve (50-x)(100-x)=88.323*100*50

$$( 50 - x ) ( 100 - x ) = 88.323 \times 100 \times 50$$

Solve for x

$$x=75-8\sqrt{6910}\approx -590.0112781$$

$$x=8\sqrt{6910}+75\approx 740.0112781$$

Steps Using the Quadratic Formula

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

$$5000-150x+x^{2}=88.323\times 100\times 50$$

Multiply 88.323 and 100 to get 8832.3.

$$5000-150x+x^{2}=8832.3\times 50$$

Multiply 8832.3 and 50 to get 441615.

$$5000-150x+x^{2}=441615$$

Subtract 441615 from both sides.

$$5000-150x+x^{2}-441615=0$$

Subtract 441615 from 5000 to get -436615.

$$-436615-150x+x^{2}=0$$

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.

$$x^{2}-150x-436615=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -150 for b, and -436615 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-150\right)±\sqrt{\left(-150\right)^{2}-4\left(-436615\right)}}{2}$$

Square -150.

$$x=\frac{-\left(-150\right)±\sqrt{22500-4\left(-436615\right)}}{2}$$

Multiply -4 times -436615.

$$x=\frac{-\left(-150\right)±\sqrt{22500+1746460}}{2}$$

Add 22500 to 1746460.

$$x=\frac{-\left(-150\right)±\sqrt{1768960}}{2}$$

Take the square root of 1768960.

$$x=\frac{-\left(-150\right)±16\sqrt{6910}}{2}$$

The opposite of -150 is 150.

$$x=\frac{150±16\sqrt{6910}}{2}$$

Now solve the equation x=\frac{150±16\sqrt{6910}}{2} when ± is plus. Add 150 to 16\sqrt{6910}.

$$x=\frac{16\sqrt{6910}+150}{2}$$

Divide 150+16\sqrt{6910} by 2.

$$x=8\sqrt{6910}+75$$

Now solve the equation x=\frac{150±16\sqrt{6910}}{2} when ± is minus. Subtract 16\sqrt{6910} from 150.

$$x=\frac{150-16\sqrt{6910}}{2}$$

Divide 150-16\sqrt{6910} by 2.

$$x=75-8\sqrt{6910}$$

The equation is now solved.

$$x=8\sqrt{6910}+75 x=75-8\sqrt{6910}$$

Steps for Completing the Square

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

$$5000-150x+x^{2}=88.323\times 100\times 50$$

Multiply 88.323 and 100 to get 8832.3.

$$5000-150x+x^{2}=8832.3\times 50$$

Multiply 8832.3 and 50 to get 441615.

$$5000-150x+x^{2}=441615$$

Subtract 5000 from both sides.

$$-150x+x^{2}=441615-5000$$

Subtract 5000 from 441615 to get 436615.

$$-150x+x^{2}=436615$$

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.

$$x^{2}-150x=436615$$

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.

$$x^{2}-150x+\left(-75\right)^{2}=436615+\left(-75\right)^{2}$$

Square -75.

$$x^{2}-150x+5625=436615+5625$$

Add 436615 to 5625.

$$x^{2}-150x+5625=442240$$

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}.

$$\left(x-75\right)^{2}=442240$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-75\right)^{2}}=\sqrt{442240}$$

Simplify.

$$x-75=8\sqrt{6910} x-75=-8\sqrt{6910}$$

Add 75 to both sides of the equation.

$$x=8\sqrt{6910}+75 x=75-8\sqrt{6910}$$