Solve 2500xleft(x+1right)=3600

$$2500x \left( x+1 \right) =3600$$

Solve for x

$$x=\frac{4}{5}=0.8$$

$$x = -\frac{9}{5} = -1\frac{4}{5} = -1.8$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2500x by x+1.

$$2500x^{2}+2500x=3600$$

Subtract 3600 from both sides.

$$2500x^{2}+2500x-3600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2500 for a, 2500 for b, and -3600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2500±\sqrt{2500^{2}-4\times 2500\left(-3600\right)}}{2\times 2500}$$

Square 2500.

$$x=\frac{-2500±\sqrt{6250000-4\times 2500\left(-3600\right)}}{2\times 2500}$$

Multiply -4 times 2500.

$$x=\frac{-2500±\sqrt{6250000-10000\left(-3600\right)}}{2\times 2500}$$

Multiply -10000 times -3600.

$$x=\frac{-2500±\sqrt{6250000+36000000}}{2\times 2500}$$

Add 6250000 to 36000000.

$$x=\frac{-2500±\sqrt{42250000}}{2\times 2500}$$

Take the square root of 42250000.

$$x=\frac{-2500±6500}{2\times 2500}$$

Multiply 2 times 2500.

$$x=\frac{-2500±6500}{5000}$$

Now solve the equation x=\frac{-2500±6500}{5000} when ± is plus. Add -2500 to 6500.

$$x=\frac{4000}{5000}$$

Reduce the fraction \frac{4000}{5000} to lowest terms by extracting and canceling out 1000.

$$x=\frac{4}{5}$$

Now solve the equation x=\frac{-2500±6500}{5000} when ± is minus. Subtract 6500 from -2500.

$$x=-\frac{9000}{5000}$$

Reduce the fraction \frac{-9000}{5000} to lowest terms by extracting and canceling out 1000.

$$x=-\frac{9}{5}$$

The equation is now solved.

$$x=\frac{4}{5} x=-\frac{9}{5}$$

Steps for Completing the Square

Use the distributive property to multiply 2500x by x+1.

$$2500x^{2}+2500x=3600$$

Divide both sides by 2500.

$$\frac{2500x^{2}+2500x}{2500}=\frac{3600}{2500}$$

Dividing by 2500 undoes the multiplication by 2500.

$$x^{2}+\frac{2500}{2500}x=\frac{3600}{2500}$$

Divide 2500 by 2500.

$$x^{2}+x=\frac{3600}{2500}$$

Reduce the fraction \frac{3600}{2500} to lowest terms by extracting and canceling out 100.

$$x^{2}+x=\frac{36}{25}$$

Divide 1, the coefficient of the x term, by 2 to get \frac{1}{2}. Then add the square of \frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=\frac{36}{25}+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+x+\frac{1}{4}=\frac{36}{25}+\frac{1}{4}$$

Add \frac{36}{25} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+x+\frac{1}{4}=\frac{169}{100}$$

Factor x^{2}+x+\frac{1}{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}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{169}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{169}{100}}$$

Simplify.

$$x+\frac{1}{2}=\frac{13}{10} x+\frac{1}{2}=-\frac{13}{10}$$

Subtract \frac{1}{2} from both sides of the equation.

$$x=\frac{4}{5} x=-\frac{9}{5}$$