Solve 3x(1+8x)=0

$$3 x ( 1 + 8 x ) = 0$$

Solve for x

$$x=-\frac{1}{8}=-0.125$$

$$x=0$$

Steps Using Factoring

Use the distributive property to multiply 3x by 1+8x.

$$3x+24x^{2}=0$$

Factor out x.

$$x\left(3+24x\right)=0$$

To find equation solutions, solve x=0 and 3+24x=0.

$$x=0 x=-\frac{1}{8}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 3x by 1+8x.

$$3x+24x^{2}=0$$

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.

$$24x^{2}+3x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 24 for a, 3 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}}}{2\times 24}$$

Take the square root of 3^{2}.

$$x=\frac{-3±3}{2\times 24}$$

Multiply 2 times 24.

$$x=\frac{-3±3}{48}$$

Now solve the equation x=\frac{-3±3}{48} when ± is plus. Add -3 to 3.

$$x=\frac{0}{48}$$

Divide 0 by 48.

$$x=0$$

Now solve the equation x=\frac{-3±3}{48} when ± is minus. Subtract 3 from -3.

$$x=-\frac{6}{48}$$

Reduce the fraction \frac{-6}{48} to lowest terms by extracting and canceling out 6.

$$x=-\frac{1}{8}$$

The equation is now solved.

$$x=0 x=-\frac{1}{8}$$

Steps for Completing the Square

Use the distributive property to multiply 3x by 1+8x.

$$3x+24x^{2}=0$$

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.

$$24x^{2}+3x=0$$

Divide both sides by 24.

$$\frac{24x^{2}+3x}{24}=\frac{0}{24}$$

Dividing by 24 undoes the multiplication by 24.

$$x^{2}+\frac{3}{24}x=\frac{0}{24}$$

Reduce the fraction \frac{3}{24} to lowest terms by extracting and canceling out 3.

$$x^{2}+\frac{1}{8}x=\frac{0}{24}$$

Divide 0 by 24.

$$x^{2}+\frac{1}{8}x=0$$

Divide \frac{1}{8}, the coefficient of the x term, by 2 to get \frac{1}{16}. Then add the square of \frac{1}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{8}x+\left(\frac{1}{16}\right)^{2}=\left(\frac{1}{16}\right)^{2}$$

Square \frac{1}{16} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{8}x+\frac{1}{256}=\frac{1}{256}$$

Factor x^{2}+\frac{1}{8}x+\frac{1}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{16}\right)^{2}=\frac{1}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{16}\right)^{2}}=\sqrt{\frac{1}{256}}$$

Simplify.

$$x+\frac{1}{16}=\frac{1}{16} x+\frac{1}{16}=-\frac{1}{16}$$

Subtract \frac{1}{16} from both sides of the equation.

$$x=0 x=-\frac{1}{8}$$