Solve 5xleft(x+1right)=684

$$5x \left( x+1 \right) = 684$$

Solve for x

$$x=\frac{\sqrt{13705}}{10}-\frac{1}{2}\approx 11.20683561$$

$$x=-\frac{\sqrt{13705}}{10}-\frac{1}{2}\approx -12.20683561$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 5x by x+1.

$$5x^{2}+5x=684$$

Subtract 684 from both sides.

$$5x^{2}+5x-684=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 5 for b, and -684 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\times 5\left(-684\right)}}{2\times 5}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\times 5\left(-684\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-5±\sqrt{25-20\left(-684\right)}}{2\times 5}$$

Multiply -20 times -684.

$$x=\frac{-5±\sqrt{25+13680}}{2\times 5}$$

Add 25 to 13680.

$$x=\frac{-5±\sqrt{13705}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{-5±\sqrt{13705}}{10}$$

Now solve the equation x=\frac{-5±\sqrt{13705}}{10} when ± is plus. Add -5 to \sqrt{13705}.

$$x=\frac{\sqrt{13705}-5}{10}$$

Divide -5+\sqrt{13705} by 10.

$$x=\frac{\sqrt{13705}}{10}-\frac{1}{2}$$

Now solve the equation x=\frac{-5±\sqrt{13705}}{10} when ± is minus. Subtract \sqrt{13705} from -5.

$$x=\frac{-\sqrt{13705}-5}{10}$$

Divide -5-\sqrt{13705} by 10.

$$x=-\frac{\sqrt{13705}}{10}-\frac{1}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{13705}}{10}-\frac{1}{2} x=-\frac{\sqrt{13705}}{10}-\frac{1}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 5x by x+1.

$$5x^{2}+5x=684$$

Divide both sides by 5.

$$\frac{5x^{2}+5x}{5}=\frac{684}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}+\frac{5}{5}x=\frac{684}{5}$$

Divide 5 by 5.

$$x^{2}+x=\frac{684}{5}$$

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.

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=\frac{684}{5}+\left(\frac{1}{2}\right)^{2}$$

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

$$x^{2}+x+\frac{1}{4}=\frac{684}{5}+\frac{1}{4}$$

Add \frac{684}{5} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+x+\frac{1}{4}=\frac{2741}{20}$$

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}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{2741}{20}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{2741}{20}}$$

Simplify.

$$x+\frac{1}{2}=\frac{\sqrt{13705}}{10} x+\frac{1}{2}=-\frac{\sqrt{13705}}{10}$$

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

$$x=\frac{\sqrt{13705}}{10}-\frac{1}{2} x=-\frac{\sqrt{13705}}{10}-\frac{1}{2}$$