Solve 5/12x=12x-15

$$\frac{ 5 }{ 12x } =12x-15$$

Solve for x

$$x=-\frac{7\sqrt{5}}{24}+\frac{5}{8}\approx -0.027186493$$

$$x=\frac{7\sqrt{5}}{24}+\frac{5}{8}\approx 1.277186493$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 12x.

$$5=12x\times 12x+12x\left(-15\right)$$

Multiply x and x to get x^{2}.

$$5=12x^{2}\times 12+12x\left(-15\right)$$

Multiply 12 and 12 to get 144.

$$5=144x^{2}+12x\left(-15\right)$$

Multiply 12 and -15 to get -180.

$$5=144x^{2}-180x$$

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

$$144x^{2}-180x=5$$

Subtract 5 from both sides.

$$144x^{2}-180x-5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 144 for a, -180 for b, and -5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-180\right)±\sqrt{\left(-180\right)^{2}-4\times 144\left(-5\right)}}{2\times 144}$$

Square -180.

$$x=\frac{-\left(-180\right)±\sqrt{32400-4\times 144\left(-5\right)}}{2\times 144}$$

Multiply -4 times 144.

$$x=\frac{-\left(-180\right)±\sqrt{32400-576\left(-5\right)}}{2\times 144}$$

Multiply -576 times -5.

$$x=\frac{-\left(-180\right)±\sqrt{32400+2880}}{2\times 144}$$

Add 32400 to 2880.

$$x=\frac{-\left(-180\right)±\sqrt{35280}}{2\times 144}$$

Take the square root of 35280.

$$x=\frac{-\left(-180\right)±84\sqrt{5}}{2\times 144}$$

The opposite of -180 is 180.

$$x=\frac{180±84\sqrt{5}}{2\times 144}$$

Multiply 2 times 144.

$$x=\frac{180±84\sqrt{5}}{288}$$

Now solve the equation x=\frac{180±84\sqrt{5}}{288} when ± is plus. Add 180 to 84\sqrt{5}.

$$x=\frac{84\sqrt{5}+180}{288}$$

Divide 180+84\sqrt{5} by 288.

$$x=\frac{7\sqrt{5}}{24}+\frac{5}{8}$$

Now solve the equation x=\frac{180±84\sqrt{5}}{288} when ± is minus. Subtract 84\sqrt{5} from 180.

$$x=\frac{180-84\sqrt{5}}{288}$$

Divide 180-84\sqrt{5} by 288.

$$x=-\frac{7\sqrt{5}}{24}+\frac{5}{8}$$

The equation is now solved.

$$x=\frac{7\sqrt{5}}{24}+\frac{5}{8} x=-\frac{7\sqrt{5}}{24}+\frac{5}{8}$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 12x.

$$5=12x\times 12x+12x\left(-15\right)$$

Multiply x and x to get x^{2}.

$$5=12x^{2}\times 12+12x\left(-15\right)$$

Multiply 12 and 12 to get 144.

$$5=144x^{2}+12x\left(-15\right)$$

Multiply 12 and -15 to get -180.

$$5=144x^{2}-180x$$

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

$$144x^{2}-180x=5$$

Divide both sides by 144.

$$\frac{144x^{2}-180x}{144}=\frac{5}{144}$$

Dividing by 144 undoes the multiplication by 144.

$$x^{2}+\left(-\frac{180}{144}\right)x=\frac{5}{144}$$

Reduce the fraction \frac{-180}{144} to lowest terms by extracting and canceling out 36.

$$x^{2}-\frac{5}{4}x=\frac{5}{144}$$

Divide -\frac{5}{4}, the coefficient of the x term, by 2 to get -\frac{5}{8}. Then add the square of -\frac{5}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{5}{4}x+\left(-\frac{5}{8}\right)^{2}=\frac{5}{144}+\left(-\frac{5}{8}\right)^{2}$$

Square -\frac{5}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{5}{4}x+\frac{25}{64}=\frac{5}{144}+\frac{25}{64}$$

Add \frac{5}{144} to \frac{25}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{5}{4}x+\frac{25}{64}=\frac{245}{576}$$

Factor x^{2}-\frac{5}{4}x+\frac{25}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5}{8}\right)^{2}=\frac{245}{576}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{8}\right)^{2}}=\sqrt{\frac{245}{576}}$$

Simplify.

$$x-\frac{5}{8}=\frac{7\sqrt{5}}{24} x-\frac{5}{8}=-\frac{7\sqrt{5}}{24}$$

Add \frac{5}{8} to both sides of the equation.

$$x=\frac{7\sqrt{5}}{24}+\frac{5}{8} x=-\frac{7\sqrt{5}}{24}+\frac{5}{8}$$