Solve 2/3(x+5)=1/4(5x-3)

$$\frac{ 2 }{ 3 } (x+5)= \frac{ 1 }{ 4(5x-3) }$$

Solve for x

$$x=\frac{\sqrt{3166}}{20}-\frac{11}{5}\approx 0.613360979$$

$$x=-\frac{\sqrt{3166}}{20}-\frac{11}{5}\approx -5.013360979$$

Steps Using the Quadratic Formula

Variable x cannot be equal to \frac{3}{5} since division by zero is not defined. Multiply both sides of the equation by 12\left(5x-3\right), the least common multiple of 3,4\left(5x-3\right).

$$\frac{2}{3}\left(x+5\right)\times 12\left(5x-3\right)=3$$

Multiply \frac{2}{3} and 12 to get 8.

$$8\left(x+5\right)\left(5x-3\right)=3$$

Use the distributive property to multiply 8 by x+5.

$$\left(8x+40\right)\left(5x-3\right)=3$$

Use the distributive property to multiply 8x+40 by 5x-3 and combine like terms.

$$40x^{2}+176x-120=3$$

Subtract 3 from both sides.

$$40x^{2}+176x-120-3=0$$

Subtract 3 from -120 to get -123.

$$40x^{2}+176x-123=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 40 for a, 176 for b, and -123 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-176±\sqrt{176^{2}-4\times 40\left(-123\right)}}{2\times 40}$$

Square 176.

$$x=\frac{-176±\sqrt{30976-4\times 40\left(-123\right)}}{2\times 40}$$

Multiply -4 times 40.

$$x=\frac{-176±\sqrt{30976-160\left(-123\right)}}{2\times 40}$$

Multiply -160 times -123.

$$x=\frac{-176±\sqrt{30976+19680}}{2\times 40}$$

Add 30976 to 19680.

$$x=\frac{-176±\sqrt{50656}}{2\times 40}$$

Take the square root of 50656.

$$x=\frac{-176±4\sqrt{3166}}{2\times 40}$$

Multiply 2 times 40.

$$x=\frac{-176±4\sqrt{3166}}{80}$$

Now solve the equation x=\frac{-176±4\sqrt{3166}}{80} when ± is plus. Add -176 to 4\sqrt{3166}.

$$x=\frac{4\sqrt{3166}-176}{80}$$

Divide -176+4\sqrt{3166} by 80.

$$x=\frac{\sqrt{3166}}{20}-\frac{11}{5}$$

Now solve the equation x=\frac{-176±4\sqrt{3166}}{80} when ± is minus. Subtract 4\sqrt{3166} from -176.

$$x=\frac{-4\sqrt{3166}-176}{80}$$

Divide -176-4\sqrt{3166} by 80.

$$x=-\frac{\sqrt{3166}}{20}-\frac{11}{5}$$

The equation is now solved.

$$x=\frac{\sqrt{3166}}{20}-\frac{11}{5} x=-\frac{\sqrt{3166}}{20}-\frac{11}{5}$$

Steps for Completing the Square

Variable x cannot be equal to \frac{3}{5} since division by zero is not defined. Multiply both sides of the equation by 12\left(5x-3\right), the least common multiple of 3,4\left(5x-3\right).

$$\frac{2}{3}\left(x+5\right)\times 12\left(5x-3\right)=3$$

Multiply \frac{2}{3} and 12 to get 8.

$$8\left(x+5\right)\left(5x-3\right)=3$$

Use the distributive property to multiply 8 by x+5.

$$\left(8x+40\right)\left(5x-3\right)=3$$

Use the distributive property to multiply 8x+40 by 5x-3 and combine like terms.

$$40x^{2}+176x-120=3$$

Add 120 to both sides.

$$40x^{2}+176x=3+120$$

Add 3 and 120 to get 123.

$$40x^{2}+176x=123$$

Divide both sides by 40.

$$\frac{40x^{2}+176x}{40}=\frac{123}{40}$$

Dividing by 40 undoes the multiplication by 40.

$$x^{2}+\frac{176}{40}x=\frac{123}{40}$$

Reduce the fraction \frac{176}{40} to lowest terms by extracting and canceling out 8.

$$x^{2}+\frac{22}{5}x=\frac{123}{40}$$

Divide \frac{22}{5}, the coefficient of the x term, by 2 to get \frac{11}{5}. Then add the square of \frac{11}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{22}{5}x+\left(\frac{11}{5}\right)^{2}=\frac{123}{40}+\left(\frac{11}{5}\right)^{2}$$

Square \frac{11}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{22}{5}x+\frac{121}{25}=\frac{123}{40}+\frac{121}{25}$$

Add \frac{123}{40} to \frac{121}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{22}{5}x+\frac{121}{25}=\frac{1583}{200}$$

Factor x^{2}+\frac{22}{5}x+\frac{121}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{11}{5}\right)^{2}=\frac{1583}{200}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{11}{5}\right)^{2}}=\sqrt{\frac{1583}{200}}$$

Simplify.

$$x+\frac{11}{5}=\frac{\sqrt{3166}}{20} x+\frac{11}{5}=-\frac{\sqrt{3166}}{20}$$

Subtract \frac{11}{5} from both sides of the equation.

$$x=\frac{\sqrt{3166}}{20}-\frac{11}{5} x=-\frac{\sqrt{3166}}{20}-\frac{11}{5}$$