Solve 3x(x+4)=12x(x+5)

$$3 x ( x + 4 ) = 12 x ( x + 5 )$$

Solve for x

$$x = -\frac{16}{3} = -5\frac{1}{3} \approx -5.333333333$$

$$x=0$$

Steps Using Factoring

Use the distributive property to multiply 3x by x+4.

$$3x^{2}+12x=12x\left(x+5\right)$$

Use the distributive property to multiply 12x by x+5.

$$3x^{2}+12x=12x^{2}+60x$$

Subtract 12x^{2} from both sides.

$$3x^{2}+12x-12x^{2}=60x$$

Combine 3x^{2} and -12x^{2} to get -9x^{2}.

$$-9x^{2}+12x=60x$$

Subtract 60x from both sides.

$$-9x^{2}+12x-60x=0$$

Combine 12x and -60x to get -48x.

$$-9x^{2}-48x=0$$

Factor out x.

$$x\left(-9x-48\right)=0$$

To find equation solutions, solve x=0 and -9x-48=0.

$$x=0 x=-\frac{16}{3}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 3x by x+4.

$$3x^{2}+12x=12x\left(x+5\right)$$

Use the distributive property to multiply 12x by x+5.

$$3x^{2}+12x=12x^{2}+60x$$

Subtract 12x^{2} from both sides.

$$3x^{2}+12x-12x^{2}=60x$$

Combine 3x^{2} and -12x^{2} to get -9x^{2}.

$$-9x^{2}+12x=60x$$

Subtract 60x from both sides.

$$-9x^{2}+12x-60x=0$$

Combine 12x and -60x to get -48x.

$$-9x^{2}-48x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, -48 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-48\right)±\sqrt{\left(-48\right)^{2}}}{2\left(-9\right)}$$

Take the square root of \left(-48\right)^{2}.

$$x=\frac{-\left(-48\right)±48}{2\left(-9\right)}$$

The opposite of -48 is 48.

$$x=\frac{48±48}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{48±48}{-18}$$

Now solve the equation x=\frac{48±48}{-18} when ± is plus. Add 48 to 48.

$$x=\frac{96}{-18}$$

Reduce the fraction \frac{96}{-18} to lowest terms by extracting and canceling out 6.

$$x=-\frac{16}{3}$$

Now solve the equation x=\frac{48±48}{-18} when ± is minus. Subtract 48 from 48.

$$x=\frac{0}{-18}$$

Divide 0 by -18.

$$x=0$$

The equation is now solved.

$$x=-\frac{16}{3} x=0$$

Steps for Completing the Square

Use the distributive property to multiply 3x by x+4.

$$3x^{2}+12x=12x\left(x+5\right)$$

Use the distributive property to multiply 12x by x+5.

$$3x^{2}+12x=12x^{2}+60x$$

Subtract 12x^{2} from both sides.

$$3x^{2}+12x-12x^{2}=60x$$

Combine 3x^{2} and -12x^{2} to get -9x^{2}.

$$-9x^{2}+12x=60x$$

Subtract 60x from both sides.

$$-9x^{2}+12x-60x=0$$

Combine 12x and -60x to get -48x.

$$-9x^{2}-48x=0$$

Divide both sides by -9.

$$\frac{-9x^{2}-48x}{-9}=\frac{0}{-9}$$

Dividing by -9 undoes the multiplication by -9.

$$x^{2}+\left(-\frac{48}{-9}\right)x=\frac{0}{-9}$$

Reduce the fraction \frac{-48}{-9} to lowest terms by extracting and canceling out 3.

$$x^{2}+\frac{16}{3}x=\frac{0}{-9}$$

Divide 0 by -9.

$$x^{2}+\frac{16}{3}x=0$$

Divide \frac{16}{3}, the coefficient of the x term, by 2 to get \frac{8}{3}. Then add the square of \frac{8}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{16}{3}x+\left(\frac{8}{3}\right)^{2}=\left(\frac{8}{3}\right)^{2}$$

Square \frac{8}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{16}{3}x+\frac{64}{9}=\frac{64}{9}$$

Factor x^{2}+\frac{16}{3}x+\frac{64}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{8}{3}\right)^{2}=\frac{64}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{8}{3}\right)^{2}}=\sqrt{\frac{64}{9}}$$

Simplify.

$$x+\frac{8}{3}=\frac{8}{3} x+\frac{8}{3}=-\frac{8}{3}$$

Subtract \frac{8}{3} from both sides of the equation.

$$x=0 x=-\frac{16}{3}$$