Solve x/x-5-3/2x=15+7/2x^2-10x

$$\frac { x } { x - 5 } - \frac { 3 } { 2 x } = \frac { 15 + 7 } { 2 x ^ { 2 } - 10 x }$$

Solve for x

$$x = \frac{\sqrt{65} + 3}{4} \approx 2.765564437$$

$$x=\frac{3-\sqrt{65}}{4}\approx -1.265564437$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,5 since division by zero is not defined. Multiply both sides of the equation by 2x\left(x-5\right), the least common multiple of x-5,2x,2x^{2}-10x.

$$2xx-\left(x-5\right)\times 3=15+7$$

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

$$2x^{2}-\left(x-5\right)\times 3=15+7$$

Use the distributive property to multiply x-5 by 3.

$$2x^{2}-\left(3x-15\right)=15+7$$

To find the opposite of 3x-15, find the opposite of each term.

$$2x^{2}-3x+15=15+7$$

Add 15 and 7 to get 22.

$$2x^{2}-3x+15=22$$

Subtract 22 from both sides.

$$2x^{2}-3x+15-22=0$$

Subtract 22 from 15 to get -7.

$$2x^{2}-3x-7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -3 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 2\left(-7\right)}}{2\times 2}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 2\left(-7\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-3\right)±\sqrt{9-8\left(-7\right)}}{2\times 2}$$

Multiply -8 times -7.

$$x=\frac{-\left(-3\right)±\sqrt{9+56}}{2\times 2}$$

Add 9 to 56.

$$x=\frac{-\left(-3\right)±\sqrt{65}}{2\times 2}$$

The opposite of -3 is 3.

$$x=\frac{3±\sqrt{65}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{3±\sqrt{65}}{4}$$

Now solve the equation x=\frac{3±\sqrt{65}}{4} when ± is plus. Add 3 to \sqrt{65}.

$$x=\frac{\sqrt{65}+3}{4}$$

Now solve the equation x=\frac{3±\sqrt{65}}{4} when ± is minus. Subtract \sqrt{65} from 3.

$$x=\frac{3-\sqrt{65}}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{65}+3}{4} x=\frac{3-\sqrt{65}}{4}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,5 since division by zero is not defined. Multiply both sides of the equation by 2x\left(x-5\right), the least common multiple of x-5,2x,2x^{2}-10x.

$$2xx-\left(x-5\right)\times 3=15+7$$

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

$$2x^{2}-\left(x-5\right)\times 3=15+7$$

Use the distributive property to multiply x-5 by 3.

$$2x^{2}-\left(3x-15\right)=15+7$$

To find the opposite of 3x-15, find the opposite of each term.

$$2x^{2}-3x+15=15+7$$

Add 15 and 7 to get 22.

$$2x^{2}-3x+15=22$$

Subtract 15 from both sides.

$$2x^{2}-3x=22-15$$

Subtract 15 from 22 to get 7.

$$2x^{2}-3x=7$$

Divide both sides by 2.

$$\frac{2x^{2}-3x}{2}=\frac{7}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}-\frac{3}{2}x=\frac{7}{2}$$

Divide -\frac{3}{2}, the coefficient of the x term, by 2 to get -\frac{3}{4}. Then add the square of -\frac{3}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{3}{2}x+\left(-\frac{3}{4}\right)^{2}=\frac{7}{2}+\left(-\frac{3}{4}\right)^{2}$$

Square -\frac{3}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{7}{2}+\frac{9}{16}$$

Add \frac{7}{2} to \frac{9}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{65}{16}$$

Factor x^{2}-\frac{3}{2}x+\frac{9}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{3}{4}\right)^{2}=\frac{65}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{4}\right)^{2}}=\sqrt{\frac{65}{16}}$$

Simplify.

$$x-\frac{3}{4}=\frac{\sqrt{65}}{4} x-\frac{3}{4}=-\frac{\sqrt{65}}{4}$$

Add \frac{3}{4} to both sides of the equation.

$$x=\frac{\sqrt{65}+3}{4} x=\frac{3-\sqrt{65}}{4}$$