Solve 4/5x-6/5=2/3x+1/3

$$\frac { 4 } { 5 } x - \frac { 6 } { 5 } = \frac { 2 } { 3 x } + \frac { 1 } { 3 }$$

Solve for x

$$x = \frac{\sqrt{1009} + 23}{24} \approx 2.281865015$$

$$x=\frac{23-\sqrt{1009}}{24}\approx -0.365198348$$

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 15x, the least common multiple of 5,3x,3.

$$\frac{4}{5}x\times 15x+15x\left(-\frac{6}{5}\right)=5\times 2+15x\times \frac{1}{3}$$

Multiply \frac{4}{5} and 15 to get 12.

$$12xx+15x\left(-\frac{6}{5}\right)=5\times 2+15x\times \frac{1}{3}$$

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

$$12x^{2}+15x\left(-\frac{6}{5}\right)=5\times 2+15x\times \frac{1}{3}$$

Multiply 15 and -\frac{6}{5} to get -18.

$$12x^{2}-18x=5\times 2+15x\times \frac{1}{3}$$

Multiply 5 and 2 to get 10.

$$12x^{2}-18x=10+15x\times \frac{1}{3}$$

Multiply 15 and \frac{1}{3} to get 5.

$$12x^{2}-18x=10+5x$$

Subtract 10 from both sides.

$$12x^{2}-18x-10=5x$$

Subtract 5x from both sides.

$$12x^{2}-18x-10-5x=0$$

Combine -18x and -5x to get -23x.

$$12x^{2}-23x-10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 12 for a, -23 for b, and -10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-23\right)±\sqrt{\left(-23\right)^{2}-4\times 12\left(-10\right)}}{2\times 12}$$

Square -23.

$$x=\frac{-\left(-23\right)±\sqrt{529-4\times 12\left(-10\right)}}{2\times 12}$$

Multiply -4 times 12.

$$x=\frac{-\left(-23\right)±\sqrt{529-48\left(-10\right)}}{2\times 12}$$

Multiply -48 times -10.

$$x=\frac{-\left(-23\right)±\sqrt{529+480}}{2\times 12}$$

Add 529 to 480.

$$x=\frac{-\left(-23\right)±\sqrt{1009}}{2\times 12}$$

The opposite of -23 is 23.

$$x=\frac{23±\sqrt{1009}}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{23±\sqrt{1009}}{24}$$

Now solve the equation x=\frac{23±\sqrt{1009}}{24} when ± is plus. Add 23 to \sqrt{1009}.

$$x=\frac{\sqrt{1009}+23}{24}$$

Now solve the equation x=\frac{23±\sqrt{1009}}{24} when ± is minus. Subtract \sqrt{1009} from 23.

$$x=\frac{23-\sqrt{1009}}{24}$$

The equation is now solved.

$$x=\frac{\sqrt{1009}+23}{24} x=\frac{23-\sqrt{1009}}{24}$$

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 15x, the least common multiple of 5,3x,3.

$$\frac{4}{5}x\times 15x+15x\left(-\frac{6}{5}\right)=5\times 2+15x\times \frac{1}{3}$$

Multiply \frac{4}{5} and 15 to get 12.

$$12xx+15x\left(-\frac{6}{5}\right)=5\times 2+15x\times \frac{1}{3}$$

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

$$12x^{2}+15x\left(-\frac{6}{5}\right)=5\times 2+15x\times \frac{1}{3}$$

Multiply 15 and -\frac{6}{5} to get -18.

$$12x^{2}-18x=5\times 2+15x\times \frac{1}{3}$$

Multiply 5 and 2 to get 10.

$$12x^{2}-18x=10+15x\times \frac{1}{3}$$

Multiply 15 and \frac{1}{3} to get 5.

$$12x^{2}-18x=10+5x$$

Subtract 5x from both sides.

$$12x^{2}-18x-5x=10$$

Combine -18x and -5x to get -23x.

$$12x^{2}-23x=10$$

Divide both sides by 12.

$$\frac{12x^{2}-23x}{12}=\frac{10}{12}$$

Dividing by 12 undoes the multiplication by 12.

$$x^{2}-\frac{23}{12}x=\frac{10}{12}$$

Reduce the fraction \frac{10}{12} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{23}{12}x=\frac{5}{6}$$

Divide -\frac{23}{12}, the coefficient of the x term, by 2 to get -\frac{23}{24}. Then add the square of -\frac{23}{24} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{23}{12}x+\left(-\frac{23}{24}\right)^{2}=\frac{5}{6}+\left(-\frac{23}{24}\right)^{2}$$

Square -\frac{23}{24} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{23}{12}x+\frac{529}{576}=\frac{5}{6}+\frac{529}{576}$$

Add \frac{5}{6} to \frac{529}{576} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{23}{12}x+\frac{529}{576}=\frac{1009}{576}$$

Factor x^{2}-\frac{23}{12}x+\frac{529}{576}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{23}{24}\right)^{2}=\frac{1009}{576}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{23}{24}\right)^{2}}=\sqrt{\frac{1009}{576}}$$

Simplify.

$$x-\frac{23}{24}=\frac{\sqrt{1009}}{24} x-\frac{23}{24}=-\frac{\sqrt{1009}}{24}$$

Add \frac{23}{24} to both sides of the equation.

$$x=\frac{\sqrt{1009}+23}{24} x=\frac{23-\sqrt{1009}}{24}$$