Solve 0=60x+8x^2

$$0 = 60 x + 8 x ^ { 2 }$$

Solve for x

$$x = -\frac{15}{2} = -7\frac{1}{2} = -7.5$$

$$x=0$$

Steps Using Factoring

Swap sides so that all variable terms are on the left hand side.

$$60x+8x^{2}=0$$

Factor out x.

$$x\left(60+8x\right)=0$$

To find equation solutions, solve x=0 and 60+8x=0.

$$x=0 x=-\frac{15}{2}$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$60x+8x^{2}=0$$

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.

$$8x^{2}+60x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, 60 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-60±\sqrt{60^{2}}}{2\times 8}$$

Take the square root of 60^{2}.

$$x=\frac{-60±60}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{-60±60}{16}$$

Now solve the equation x=\frac{-60±60}{16} when ± is plus. Add -60 to 60.

$$x=\frac{0}{16}$$

Divide 0 by 16.

$$x=0$$

Now solve the equation x=\frac{-60±60}{16} when ± is minus. Subtract 60 from -60.

$$x=-\frac{120}{16}$$

Reduce the fraction \frac{-120}{16} to lowest terms by extracting and canceling out 8.

$$x=-\frac{15}{2}$$

The equation is now solved.

$$x=0 x=-\frac{15}{2}$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$60x+8x^{2}=0$$

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.

$$8x^{2}+60x=0$$

Divide both sides by 8.

$$\frac{8x^{2}+60x}{8}=\frac{0}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$x^{2}+\frac{60}{8}x=\frac{0}{8}$$

Reduce the fraction \frac{60}{8} to lowest terms by extracting and canceling out 4.

$$x^{2}+\frac{15}{2}x=\frac{0}{8}$$

Divide 0 by 8.

$$x^{2}+\frac{15}{2}x=0$$

Divide \frac{15}{2}, the coefficient of the x term, by 2 to get \frac{15}{4}. Then add the square of \frac{15}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{15}{2}x+\left(\frac{15}{4}\right)^{2}=\left(\frac{15}{4}\right)^{2}$$

Square \frac{15}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{15}{2}x+\frac{225}{16}=\frac{225}{16}$$

Factor x^{2}+\frac{15}{2}x+\frac{225}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{15}{4}\right)^{2}=\frac{225}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{15}{4}\right)^{2}}=\sqrt{\frac{225}{16}}$$

Simplify.

$$x+\frac{15}{4}=\frac{15}{4} x+\frac{15}{4}=-\frac{15}{4}$$

Subtract \frac{15}{4} from both sides of the equation.

$$x=0 x=-\frac{15}{2}$$