Solve 100(x+50+x)(x+150+50)=306600

$$100(x+50+x)(x+150+50)=306600$$

Solve for x

$$x=\frac{\sqrt{36757}-225}{2}\approx -16.639424162$$

$$x=\frac{-\sqrt{36757}-225}{2}\approx -208.360575838$$

Steps Using the Quadratic Formula

Combine x and x to get 2x.

$$100\left(2x+50\right)\left(x+150+50\right)=306600$$

Add 150 and 50 to get 200.

$$100\left(2x+50\right)\left(x+200\right)=306600$$

Use the distributive property to multiply 100 by 2x+50.

$$\left(200x+5000\right)\left(x+200\right)=306600$$

Use the distributive property to multiply 200x+5000 by x+200 and combine like terms.

$$200x^{2}+45000x+1000000=306600$$

Subtract 306600 from both sides.

$$200x^{2}+45000x+1000000-306600=0$$

Subtract 306600 from 1000000 to get 693400.

$$200x^{2}+45000x+693400=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 200 for a, 45000 for b, and 693400 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-45000±\sqrt{45000^{2}-4\times 200\times 693400}}{2\times 200}$$

Square 45000.

$$x=\frac{-45000±\sqrt{2025000000-4\times 200\times 693400}}{2\times 200}$$

Multiply -4 times 200.

$$x=\frac{-45000±\sqrt{2025000000-800\times 693400}}{2\times 200}$$

Multiply -800 times 693400.

$$x=\frac{-45000±\sqrt{2025000000-554720000}}{2\times 200}$$

Add 2025000000 to -554720000.

$$x=\frac{-45000±\sqrt{1470280000}}{2\times 200}$$

Take the square root of 1470280000.

$$x=\frac{-45000±200\sqrt{36757}}{2\times 200}$$

Multiply 2 times 200.

$$x=\frac{-45000±200\sqrt{36757}}{400}$$

Now solve the equation x=\frac{-45000±200\sqrt{36757}}{400} when ± is plus. Add -45000 to 200\sqrt{36757}.

$$x=\frac{200\sqrt{36757}-45000}{400}$$

Divide -45000+200\sqrt{36757} by 400.

$$x=\frac{\sqrt{36757}-225}{2}$$

Now solve the equation x=\frac{-45000±200\sqrt{36757}}{400} when ± is minus. Subtract 200\sqrt{36757} from -45000.

$$x=\frac{-200\sqrt{36757}-45000}{400}$$

Divide -45000-200\sqrt{36757} by 400.

$$x=\frac{-\sqrt{36757}-225}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{36757}-225}{2} x=\frac{-\sqrt{36757}-225}{2}$$

Steps for Completing the Square

Combine x and x to get 2x.

$$100\left(2x+50\right)\left(x+150+50\right)=306600$$

Add 150 and 50 to get 200.

$$100\left(2x+50\right)\left(x+200\right)=306600$$

Use the distributive property to multiply 100 by 2x+50.

$$\left(200x+5000\right)\left(x+200\right)=306600$$

Use the distributive property to multiply 200x+5000 by x+200 and combine like terms.

$$200x^{2}+45000x+1000000=306600$$

Subtract 1000000 from both sides.

$$200x^{2}+45000x=306600-1000000$$

Subtract 1000000 from 306600 to get -693400.

$$200x^{2}+45000x=-693400$$

Divide both sides by 200.

$$\frac{200x^{2}+45000x}{200}=-\frac{693400}{200}$$

Dividing by 200 undoes the multiplication by 200.

$$x^{2}+\frac{45000}{200}x=-\frac{693400}{200}$$

Divide 45000 by 200.

$$x^{2}+225x=-\frac{693400}{200}$$

Divide -693400 by 200.

$$x^{2}+225x=-3467$$

Divide 225, the coefficient of the x term, by 2 to get \frac{225}{2}. Then add the square of \frac{225}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+225x+\left(\frac{225}{2}\right)^{2}=-3467+\left(\frac{225}{2}\right)^{2}$$

Square \frac{225}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+225x+\frac{50625}{4}=-3467+\frac{50625}{4}$$

Add -3467 to \frac{50625}{4}.

$$x^{2}+225x+\frac{50625}{4}=\frac{36757}{4}$$

Factor x^{2}+225x+\frac{50625}{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}.

$$\left(x+\frac{225}{2}\right)^{2}=\frac{36757}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{225}{2}\right)^{2}}=\sqrt{\frac{36757}{4}}$$

Simplify.

$$x+\frac{225}{2}=\frac{\sqrt{36757}}{2} x+\frac{225}{2}=-\frac{\sqrt{36757}}{2}$$

Subtract \frac{225}{2} from both sides of the equation.

$$x=\frac{\sqrt{36757}-225}{2} x=\frac{-\sqrt{36757}-225}{2}$$