Solve 24/x+24x-4=60

$$\frac { 24 } { x } + 24 x - 4 = 60$$

Solve for x

$$x = \frac{\sqrt{7} + 4}{3} \approx 2.215250437$$

$$x=\frac{4-\sqrt{7}}{3}\approx 0.45141623$$

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 x.

$$24+24xx+x\left(-4\right)=60x$$

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

$$24+24x^{2}+x\left(-4\right)=60x$$

Subtract 60x from both sides.

$$24+24x^{2}+x\left(-4\right)-60x=0$$

Combine x\left(-4\right) and -60x to get -64x.

$$24+24x^{2}-64x=0$$

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.

$$24x^{2}-64x+24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 24 for a, -64 for b, and 24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-64\right)±\sqrt{\left(-64\right)^{2}-4\times 24\times 24}}{2\times 24}$$

Square -64.

$$x=\frac{-\left(-64\right)±\sqrt{4096-4\times 24\times 24}}{2\times 24}$$

Multiply -4 times 24.

$$x=\frac{-\left(-64\right)±\sqrt{4096-96\times 24}}{2\times 24}$$

Multiply -96 times 24.

$$x=\frac{-\left(-64\right)±\sqrt{4096-2304}}{2\times 24}$$

Add 4096 to -2304.

$$x=\frac{-\left(-64\right)±\sqrt{1792}}{2\times 24}$$

Take the square root of 1792.

$$x=\frac{-\left(-64\right)±16\sqrt{7}}{2\times 24}$$

The opposite of -64 is 64.

$$x=\frac{64±16\sqrt{7}}{2\times 24}$$

Multiply 2 times 24.

$$x=\frac{64±16\sqrt{7}}{48}$$

Now solve the equation x=\frac{64±16\sqrt{7}}{48} when ± is plus. Add 64 to 16\sqrt{7}.

$$x=\frac{16\sqrt{7}+64}{48}$$

Divide 64+16\sqrt{7} by 48.

$$x=\frac{\sqrt{7}+4}{3}$$

Now solve the equation x=\frac{64±16\sqrt{7}}{48} when ± is minus. Subtract 16\sqrt{7} from 64.

$$x=\frac{64-16\sqrt{7}}{48}$$

Divide 64-16\sqrt{7} by 48.

$$x=\frac{4-\sqrt{7}}{3}$$

The equation is now solved.

$$x=\frac{\sqrt{7}+4}{3} x=\frac{4-\sqrt{7}}{3}$$

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 x.

$$24+24xx+x\left(-4\right)=60x$$

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

$$24+24x^{2}+x\left(-4\right)=60x$$

Subtract 60x from both sides.

$$24+24x^{2}+x\left(-4\right)-60x=0$$

Combine x\left(-4\right) and -60x to get -64x.

$$24+24x^{2}-64x=0$$

Subtract 24 from both sides. Anything subtracted from zero gives its negation.

$$24x^{2}-64x=-24$$

Divide both sides by 24.

$$\frac{24x^{2}-64x}{24}=-\frac{24}{24}$$

Dividing by 24 undoes the multiplication by 24.

$$x^{2}+\left(-\frac{64}{24}\right)x=-\frac{24}{24}$$

Reduce the fraction \frac{-64}{24} to lowest terms by extracting and canceling out 8.

$$x^{2}-\frac{8}{3}x=-\frac{24}{24}$$

Divide -24 by 24.

$$x^{2}-\frac{8}{3}x=-1$$

Divide -\frac{8}{3}, the coefficient of the x term, by 2 to get -\frac{4}{3}. Then add the square of -\frac{4}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{8}{3}x+\left(-\frac{4}{3}\right)^{2}=-1+\left(-\frac{4}{3}\right)^{2}$$

Square -\frac{4}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{8}{3}x+\frac{16}{9}=-1+\frac{16}{9}$$

Add -1 to \frac{16}{9}.

$$x^{2}-\frac{8}{3}x+\frac{16}{9}=\frac{7}{9}$$

Factor x^{2}-\frac{8}{3}x+\frac{16}{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}.

$$\left(x-\frac{4}{3}\right)^{2}=\frac{7}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{4}{3}\right)^{2}}=\sqrt{\frac{7}{9}}$$

Simplify.

$$x-\frac{4}{3}=\frac{\sqrt{7}}{3} x-\frac{4}{3}=-\frac{\sqrt{7}}{3}$$

Add \frac{4}{3} to both sides of the equation.

$$x=\frac{\sqrt{7}+4}{3} x=\frac{4-\sqrt{7}}{3}$$