Solve 5/6x+5+x/5=20/24x+20

$$\frac { 5 } { 6 x + 5 } + \frac { x } { 5 } = \frac { 20 } { 24 x + 20 }$$

Solve for x

$$x=0$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -\frac{5}{6} since division by zero is not defined. Multiply both sides of the equation by 20\left(6x+5\right), the least common multiple of 6x+5,5,24x+20.

$$20\times 5+\left(24x+20\right)x=5\times 20$$

Multiply 20 and 5 to get 100.

$$100+\left(24x+20\right)x=5\times 20$$

Use the distributive property to multiply 24x+20 by x.

$$100+24x^{2}+20x=5\times 20$$

Multiply 5 and 20 to get 100.

$$100+24x^{2}+20x=100$$

Subtract 100 from both sides.

$$100+24x^{2}+20x-100=0$$

Subtract 100 from 100 to get 0.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 24 for a, 20 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-20±\sqrt{20^{2}}}{2\times 24}$$

Take the square root of 20^{2}.

$$x=\frac{-20±20}{2\times 24}$$

Multiply 2 times 24.

$$x=\frac{-20±20}{48}$$

Now solve the equation x=\frac{-20±20}{48} when ± is plus. Add -20 to 20.

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

Divide 0 by 48.

$$x=0$$

Now solve the equation x=\frac{-20±20}{48} when ± is minus. Subtract 20 from -20.

$$x=-\frac{40}{48}$$

Reduce the fraction \frac{-40}{48} to lowest terms by extracting and canceling out 8.

$$x=-\frac{5}{6}$$

The equation is now solved.

$$x=0 x=-\frac{5}{6}$$

Variable x cannot be equal to -\frac{5}{6}.

$$x=0$$

Steps for Completing the Square

Variable x cannot be equal to -\frac{5}{6} since division by zero is not defined. Multiply both sides of the equation by 20\left(6x+5\right), the least common multiple of 6x+5,5,24x+20.

$$20\times 5+\left(24x+20\right)x=5\times 20$$

Multiply 20 and 5 to get 100.

$$100+\left(24x+20\right)x=5\times 20$$

Use the distributive property to multiply 24x+20 by x.

$$100+24x^{2}+20x=5\times 20$$

Multiply 5 and 20 to get 100.

$$100+24x^{2}+20x=100$$

Subtract 100 from both sides.

$$24x^{2}+20x=100-100$$

Subtract 100 from 100 to get 0.

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

Divide both sides by 24.

$$\frac{24x^{2}+20x}{24}=\frac{0}{24}$$

Dividing by 24 undoes the multiplication by 24.

$$x^{2}+\frac{20}{24}x=\frac{0}{24}$$

Reduce the fraction \frac{20}{24} to lowest terms by extracting and canceling out 4.

$$x^{2}+\frac{5}{6}x=\frac{0}{24}$$

Divide 0 by 24.

$$x^{2}+\frac{5}{6}x=0$$

Divide \frac{5}{6}, the coefficient of the x term, by 2 to get \frac{5}{12}. Then add the square of \frac{5}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{5}{6}x+\left(\frac{5}{12}\right)^{2}=\left(\frac{5}{12}\right)^{2}$$

Square \frac{5}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{5}{6}x+\frac{25}{144}=\frac{25}{144}$$

Factor x^{2}+\frac{5}{6}x+\frac{25}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{5}{12}\right)^{2}=\frac{25}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{12}\right)^{2}}=\sqrt{\frac{25}{144}}$$

Simplify.

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

Subtract \frac{5}{12} from both sides of the equation.

$$x=0 x=-\frac{5}{6}$$

Variable x cannot be equal to -\frac{5}{6}.

$$x=0$$