Solve 4:2/x-4/5=3/x

$$4 : \frac { 2 } { x } - \frac { 4 } { 5 } = \frac { 3 } { x }$$

Solve for x

$$x=-\frac{\sqrt{154}}{10}+\frac{1}{5}\approx -1.040967365$$

$$x=\frac{\sqrt{154}}{10}+\frac{1}{5}\approx 1.440967365$$

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

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

Multiply \frac{5}{2} and 4 to get 10.

$$10x^{2}+5x\left(-\frac{4}{5}\right)=5\times 3$$

Multiply 5 and -\frac{4}{5} to get -4.

$$10x^{2}-4x=5\times 3$$

Multiply 5 and 3 to get 15.

$$10x^{2}-4x=15$$

Subtract 15 from both sides.

$$10x^{2}-4x-15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 10 for a, -4 for b, and -15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 10\left(-15\right)}}{2\times 10}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 10\left(-15\right)}}{2\times 10}$$

Multiply -4 times 10.

$$x=\frac{-\left(-4\right)±\sqrt{16-40\left(-15\right)}}{2\times 10}$$

Multiply -40 times -15.

$$x=\frac{-\left(-4\right)±\sqrt{16+600}}{2\times 10}$$

Add 16 to 600.

$$x=\frac{-\left(-4\right)±\sqrt{616}}{2\times 10}$$

Take the square root of 616.

$$x=\frac{-\left(-4\right)±2\sqrt{154}}{2\times 10}$$

The opposite of -4 is 4.

$$x=\frac{4±2\sqrt{154}}{2\times 10}$$

Multiply 2 times 10.

$$x=\frac{4±2\sqrt{154}}{20}$$

Now solve the equation x=\frac{4±2\sqrt{154}}{20} when ± is plus. Add 4 to 2\sqrt{154}.

$$x=\frac{2\sqrt{154}+4}{20}$$

Divide 4+2\sqrt{154} by 20.

$$x=\frac{\sqrt{154}}{10}+\frac{1}{5}$$

Now solve the equation x=\frac{4±2\sqrt{154}}{20} when ± is minus. Subtract 2\sqrt{154} from 4.

$$x=\frac{4-2\sqrt{154}}{20}$$

Divide 4-2\sqrt{154} by 20.

$$x=-\frac{\sqrt{154}}{10}+\frac{1}{5}$$

The equation is now solved.

$$x=\frac{\sqrt{154}}{10}+\frac{1}{5} x=-\frac{\sqrt{154}}{10}+\frac{1}{5}$$

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

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

Multiply \frac{5}{2} and 4 to get 10.

$$10x^{2}+5x\left(-\frac{4}{5}\right)=5\times 3$$

Multiply 5 and -\frac{4}{5} to get -4.

$$10x^{2}-4x=5\times 3$$

Multiply 5 and 3 to get 15.

$$10x^{2}-4x=15$$

Divide both sides by 10.

$$\frac{10x^{2}-4x}{10}=\frac{15}{10}$$

Dividing by 10 undoes the multiplication by 10.

$$x^{2}+\left(-\frac{4}{10}\right)x=\frac{15}{10}$$

Reduce the fraction \frac{-4}{10} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{2}{5}x=\frac{15}{10}$$

Reduce the fraction \frac{15}{10} to lowest terms by extracting and canceling out 5.

$$x^{2}-\frac{2}{5}x=\frac{3}{2}$$

Divide -\frac{2}{5}, the coefficient of the x term, by 2 to get -\frac{1}{5}. Then add the square of -\frac{1}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{1}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{2}{5}x+\frac{1}{25}=\frac{3}{2}+\frac{1}{25}$$

Add \frac{3}{2} to \frac{1}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{2}{5}x+\frac{1}{25}=\frac{77}{50}$$

Factor x^{2}-\frac{2}{5}x+\frac{1}{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}.

$$\left(x-\frac{1}{5}\right)^{2}=\frac{77}{50}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{5}\right)^{2}}=\sqrt{\frac{77}{50}}$$

Simplify.

$$x-\frac{1}{5}=\frac{\sqrt{154}}{10} x-\frac{1}{5}=-\frac{\sqrt{154}}{10}$$

Add \frac{1}{5} to both sides of the equation.

$$x=\frac{\sqrt{154}}{10}+\frac{1}{5} x=-\frac{\sqrt{154}}{10}+\frac{1}{5}$$