Solve 800x+500x(9+x)=6000

$$800x+500x(9+x)=6000$$

Solve for x

$$x = \frac{\sqrt{4009} - 53}{10} \approx 1.031666447$$

$$x=\frac{-\sqrt{4009}-53}{10}\approx -11.631666447$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 500x by 9+x.

$$800x+4500x+500x^{2}=6000$$

Combine 800x and 4500x to get 5300x.

$$5300x+500x^{2}=6000$$

Subtract 6000 from both sides.

$$5300x+500x^{2}-6000=0$$

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.

$$500x^{2}+5300x-6000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 500 for a, 5300 for b, and -6000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5300±\sqrt{5300^{2}-4\times 500\left(-6000\right)}}{2\times 500}$$

Square 5300.

$$x=\frac{-5300±\sqrt{28090000-4\times 500\left(-6000\right)}}{2\times 500}$$

Multiply -4 times 500.

$$x=\frac{-5300±\sqrt{28090000-2000\left(-6000\right)}}{2\times 500}$$

Multiply -2000 times -6000.

$$x=\frac{-5300±\sqrt{28090000+12000000}}{2\times 500}$$

Add 28090000 to 12000000.

$$x=\frac{-5300±\sqrt{40090000}}{2\times 500}$$

Take the square root of 40090000.

$$x=\frac{-5300±100\sqrt{4009}}{2\times 500}$$

Multiply 2 times 500.

$$x=\frac{-5300±100\sqrt{4009}}{1000}$$

Now solve the equation x=\frac{-5300±100\sqrt{4009}}{1000} when ± is plus. Add -5300 to 100\sqrt{4009}.

$$x=\frac{100\sqrt{4009}-5300}{1000}$$

Divide -5300+100\sqrt{4009} by 1000.

$$x=\frac{\sqrt{4009}-53}{10}$$

Now solve the equation x=\frac{-5300±100\sqrt{4009}}{1000} when ± is minus. Subtract 100\sqrt{4009} from -5300.

$$x=\frac{-100\sqrt{4009}-5300}{1000}$$

Divide -5300-100\sqrt{4009} by 1000.

$$x=\frac{-\sqrt{4009}-53}{10}$$

The equation is now solved.

$$x=\frac{\sqrt{4009}-53}{10} x=\frac{-\sqrt{4009}-53}{10}$$

Steps for Completing the Square

Use the distributive property to multiply 500x by 9+x.

$$800x+4500x+500x^{2}=6000$$

Combine 800x and 4500x to get 5300x.

$$5300x+500x^{2}=6000$$

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.

$$500x^{2}+5300x=6000$$

Divide both sides by 500.

$$\frac{500x^{2}+5300x}{500}=\frac{6000}{500}$$

Dividing by 500 undoes the multiplication by 500.

$$x^{2}+\frac{5300}{500}x=\frac{6000}{500}$$

Reduce the fraction \frac{5300}{500} to lowest terms by extracting and canceling out 100.

$$x^{2}+\frac{53}{5}x=\frac{6000}{500}$$

Divide 6000 by 500.

$$x^{2}+\frac{53}{5}x=12$$

Divide \frac{53}{5}, the coefficient of the x term, by 2 to get \frac{53}{10}. Then add the square of \frac{53}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{53}{5}x+\left(\frac{53}{10}\right)^{2}=12+\left(\frac{53}{10}\right)^{2}$$

Square \frac{53}{10} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{53}{5}x+\frac{2809}{100}=12+\frac{2809}{100}$$

Add 12 to \frac{2809}{100}.

$$x^{2}+\frac{53}{5}x+\frac{2809}{100}=\frac{4009}{100}$$

Factor x^{2}+\frac{53}{5}x+\frac{2809}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{53}{10}\right)^{2}=\frac{4009}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{53}{10}\right)^{2}}=\sqrt{\frac{4009}{100}}$$

Simplify.

$$x+\frac{53}{10}=\frac{\sqrt{4009}}{10} x+\frac{53}{10}=-\frac{\sqrt{4009}}{10}$$

Subtract \frac{53}{10} from both sides of the equation.

$$x=\frac{\sqrt{4009}-53}{10} x=\frac{-\sqrt{4009}-53}{10}$$