Solve 32(1-x)(50-x)=1156

$$32(1-x)(50-x)=1156$$

Solve for x

$$x=-\frac{\sqrt{10182}}{4}+\frac{51}{2}\approx 0.273525811$$

$$x=\frac{\sqrt{10182}}{4}+\frac{51}{2}\approx 50.726474189$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 32 by 1-x.

$$\left(32-32x\right)\left(50-x\right)=1156$$

Use the distributive property to multiply 32-32x by 50-x and combine like terms.

$$1600-1632x+32x^{2}=1156$$

Subtract 1156 from both sides.

$$1600-1632x+32x^{2}-1156=0$$

Subtract 1156 from 1600 to get 444.

$$444-1632x+32x^{2}=0$$

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.

$$32x^{2}-1632x+444=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 32 for a, -1632 for b, and 444 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1632\right)±\sqrt{\left(-1632\right)^{2}-4\times 32\times 444}}{2\times 32}$$

Square -1632.

$$x=\frac{-\left(-1632\right)±\sqrt{2663424-4\times 32\times 444}}{2\times 32}$$

Multiply -4 times 32.

$$x=\frac{-\left(-1632\right)±\sqrt{2663424-128\times 444}}{2\times 32}$$

Multiply -128 times 444.

$$x=\frac{-\left(-1632\right)±\sqrt{2663424-56832}}{2\times 32}$$

Add 2663424 to -56832.

$$x=\frac{-\left(-1632\right)±\sqrt{2606592}}{2\times 32}$$

Take the square root of 2606592.

$$x=\frac{-\left(-1632\right)±16\sqrt{10182}}{2\times 32}$$

The opposite of -1632 is 1632.

$$x=\frac{1632±16\sqrt{10182}}{2\times 32}$$

Multiply 2 times 32.

$$x=\frac{1632±16\sqrt{10182}}{64}$$

Now solve the equation x=\frac{1632±16\sqrt{10182}}{64} when ± is plus. Add 1632 to 16\sqrt{10182}.

$$x=\frac{16\sqrt{10182}+1632}{64}$$

Divide 1632+16\sqrt{10182} by 64.

$$x=\frac{\sqrt{10182}}{4}+\frac{51}{2}$$

Now solve the equation x=\frac{1632±16\sqrt{10182}}{64} when ± is minus. Subtract 16\sqrt{10182} from 1632.

$$x=\frac{1632-16\sqrt{10182}}{64}$$

Divide 1632-16\sqrt{10182} by 64.

$$x=-\frac{\sqrt{10182}}{4}+\frac{51}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{10182}}{4}+\frac{51}{2} x=-\frac{\sqrt{10182}}{4}+\frac{51}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 32 by 1-x.

$$\left(32-32x\right)\left(50-x\right)=1156$$

Use the distributive property to multiply 32-32x by 50-x and combine like terms.

$$1600-1632x+32x^{2}=1156$$

Subtract 1600 from both sides.

$$-1632x+32x^{2}=1156-1600$$

Subtract 1600 from 1156 to get -444.

$$-1632x+32x^{2}=-444$$

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.

$$32x^{2}-1632x=-444$$

Divide both sides by 32.

$$\frac{32x^{2}-1632x}{32}=-\frac{444}{32}$$

Dividing by 32 undoes the multiplication by 32.

$$x^{2}+\left(-\frac{1632}{32}\right)x=-\frac{444}{32}$$

Divide -1632 by 32.

$$x^{2}-51x=-\frac{444}{32}$$

Reduce the fraction \frac{-444}{32} to lowest terms by extracting and canceling out 4.

$$x^{2}-51x=-\frac{111}{8}$$

Divide -51, the coefficient of the x term, by 2 to get -\frac{51}{2}. Then add the square of -\frac{51}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-51x+\left(-\frac{51}{2}\right)^{2}=-\frac{111}{8}+\left(-\frac{51}{2}\right)^{2}$$

Square -\frac{51}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-51x+\frac{2601}{4}=-\frac{111}{8}+\frac{2601}{4}$$

Add -\frac{111}{8} to \frac{2601}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-51x+\frac{2601}{4}=\frac{5091}{8}$$

Factor x^{2}-51x+\frac{2601}{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}.

$$\left(x-\frac{51}{2}\right)^{2}=\frac{5091}{8}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{51}{2}\right)^{2}}=\sqrt{\frac{5091}{8}}$$

Simplify.

$$x-\frac{51}{2}=\frac{\sqrt{10182}}{4} x-\frac{51}{2}=-\frac{\sqrt{10182}}{4}$$

Add \frac{51}{2} to both sides of the equation.

$$x=\frac{\sqrt{10182}}{4}+\frac{51}{2} x=-\frac{\sqrt{10182}}{4}+\frac{51}{2}$$