Solve 5/3+(24-8x)/12x=2/3

$$5 / 3 + ( 24 - 8 x ) / 12 x = 2 / 3$$

Solve for x

$$x = \frac{\sqrt{15} + 3}{2} \approx 3.436491673$$

$$x=\frac{3-\sqrt{15}}{2}\approx -0.436491673$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 12, the least common multiple of 3,12.

$$20+\left(24-8x\right)x=8$$

Use the distributive property to multiply 24-8x by x.

$$20+24x-8x^{2}=8$$

Subtract 8 from both sides.

$$20+24x-8x^{2}-8=0$$

Subtract 8 from 20 to get 12.

$$12+24x-8x^{2}=0$$

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.

$$-8x^{2}+24x+12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -8 for a, 24 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-24±\sqrt{24^{2}-4\left(-8\right)\times 12}}{2\left(-8\right)}$$

Square 24.

$$x=\frac{-24±\sqrt{576-4\left(-8\right)\times 12}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$x=\frac{-24±\sqrt{576+32\times 12}}{2\left(-8\right)}$$

Multiply 32 times 12.

$$x=\frac{-24±\sqrt{576+384}}{2\left(-8\right)}$$

Add 576 to 384.

$$x=\frac{-24±\sqrt{960}}{2\left(-8\right)}$$

Take the square root of 960.

$$x=\frac{-24±8\sqrt{15}}{2\left(-8\right)}$$

Multiply 2 times -8.

$$x=\frac{-24±8\sqrt{15}}{-16}$$

Now solve the equation x=\frac{-24±8\sqrt{15}}{-16} when ± is plus. Add -24 to 8\sqrt{15}.

$$x=\frac{8\sqrt{15}-24}{-16}$$

Divide -24+8\sqrt{15} by -16.

$$x=\frac{3-\sqrt{15}}{2}$$

Now solve the equation x=\frac{-24±8\sqrt{15}}{-16} when ± is minus. Subtract 8\sqrt{15} from -24.

$$x=\frac{-8\sqrt{15}-24}{-16}$$

Divide -24-8\sqrt{15} by -16.

$$x=\frac{\sqrt{15}+3}{2}$$

The equation is now solved.

$$x=\frac{3-\sqrt{15}}{2} x=\frac{\sqrt{15}+3}{2}$$

Steps for Completing the Square

Multiply both sides of the equation by 12, the least common multiple of 3,12.

$$20+\left(24-8x\right)x=8$$

Use the distributive property to multiply 24-8x by x.

$$20+24x-8x^{2}=8$$

Subtract 20 from both sides.

$$24x-8x^{2}=8-20$$

Subtract 20 from 8 to get -12.

$$24x-8x^{2}=-12$$

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.

$$-8x^{2}+24x=-12$$

Divide both sides by -8.

$$\frac{-8x^{2}+24x}{-8}=-\frac{12}{-8}$$

Dividing by -8 undoes the multiplication by -8.

$$x^{2}+\frac{24}{-8}x=-\frac{12}{-8}$$

Divide 24 by -8.

$$x^{2}-3x=-\frac{12}{-8}$$

Reduce the fraction \frac{-12}{-8} to lowest terms by extracting and canceling out 4.

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

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

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

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-3x+\frac{9}{4}=\frac{3}{2}+\frac{9}{4}$$

Add \frac{3}{2} to \frac{9}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-3x+\frac{9}{4}=\frac{15}{4}$$

Factor x^{2}-3x+\frac{9}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{15}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{15}{4}}$$

Simplify.

$$x-\frac{3}{2}=\frac{\sqrt{15}}{2} x-\frac{3}{2}=-\frac{\sqrt{15}}{2}$$

Add \frac{3}{2} to both sides of the equation.

$$x=\frac{\sqrt{15}+3}{2} x=\frac{3-\sqrt{15}}{2}$$