Solve 5x/2=7x-4/x

$$\frac { 5 x } { 2 } = \frac { 7 x - 4 } { x }$$

Solve for x

$$x=2$$

$$x=\frac{4}{5}=0.8$$

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 2x, the least common multiple of 2,x.

$$x\times 5x=2\left(7x-4\right)$$

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

$$x^{2}\times 5=2\left(7x-4\right)$$

Use the distributive property to multiply 2 by 7x-4.

$$x^{2}\times 5=14x-8$$

Subtract 14x from both sides.

$$x^{2}\times 5-14x=-8$$

Add 8 to both sides.

$$x^{2}\times 5-14x+8=0$$

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.

$$5x^{2}-14x+8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, -14 for b, and 8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 5\times 8}}{2\times 5}$$

Square -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 5\times 8}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-\left(-14\right)±\sqrt{196-20\times 8}}{2\times 5}$$

Multiply -20 times 8.

$$x=\frac{-\left(-14\right)±\sqrt{196-160}}{2\times 5}$$

Add 196 to -160.

$$x=\frac{-\left(-14\right)±\sqrt{36}}{2\times 5}$$

Take the square root of 36.

$$x=\frac{-\left(-14\right)±6}{2\times 5}$$

The opposite of -14 is 14.

$$x=\frac{14±6}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{14±6}{10}$$

Now solve the equation x=\frac{14±6}{10} when ± is plus. Add 14 to 6.

$$x=\frac{20}{10}$$

Divide 20 by 10.

$$x=2$$

Now solve the equation x=\frac{14±6}{10} when ± is minus. Subtract 6 from 14.

$$x=\frac{8}{10}$$

Reduce the fraction \frac{8}{10} to lowest terms by extracting and canceling out 2.

$$x=\frac{4}{5}$$

The equation is now solved.

$$x=2 x=\frac{4}{5}$$

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 2x, the least common multiple of 2,x.

$$x\times 5x=2\left(7x-4\right)$$

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

$$x^{2}\times 5=2\left(7x-4\right)$$

Use the distributive property to multiply 2 by 7x-4.

$$x^{2}\times 5=14x-8$$

Subtract 14x from both sides.

$$x^{2}\times 5-14x=-8$$

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.

$$5x^{2}-14x=-8$$

Divide both sides by 5.

$$\frac{5x^{2}-14x}{5}=-\frac{8}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}-\frac{14}{5}x=-\frac{8}{5}$$

Divide -\frac{14}{5}, the coefficient of the x term, by 2 to get -\frac{7}{5}. Then add the square of -\frac{7}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{14}{5}x+\left(-\frac{7}{5}\right)^{2}=-\frac{8}{5}+\left(-\frac{7}{5}\right)^{2}$$

Square -\frac{7}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{14}{5}x+\frac{49}{25}=-\frac{8}{5}+\frac{49}{25}$$

Add -\frac{8}{5} to \frac{49}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{14}{5}x+\frac{49}{25}=\frac{9}{25}$$

Factor x^{2}-\frac{14}{5}x+\frac{49}{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}.

$$\left(x-\frac{7}{5}\right)^{2}=\frac{9}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{5}\right)^{2}}=\sqrt{\frac{9}{25}}$$

Simplify.

$$x-\frac{7}{5}=\frac{3}{5} x-\frac{7}{5}=-\frac{3}{5}$$

Add \frac{7}{5} to both sides of the equation.

$$x=2 x=\frac{4}{5}$$