Solve 5/x-2=7-10/x+2

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

Solve for x

$$x=-\frac{6}{7}\approx -0.857142857$$

$$x=3$$

Steps Using the Quadratic Formula

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

$$\left(x+2\right)\times 5=\left(x-2\right)\left(x+2\right)\times 7-\left(x-2\right)\times 10$$

Use the distributive property to multiply x+2 by 5.

$$5x+10=\left(x-2\right)\left(x+2\right)\times 7-\left(x-2\right)\times 10$$

Use the distributive property to multiply x-2 by x+2 and combine like terms.

$$5x+10=\left(x^{2}-4\right)\times 7-\left(x-2\right)\times 10$$

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

$$5x+10=7x^{2}-28-\left(x-2\right)\times 10$$

Use the distributive property to multiply x-2 by 10.

$$5x+10=7x^{2}-28-\left(10x-20\right)$$

To find the opposite of 10x-20, find the opposite of each term.

$$5x+10=7x^{2}-28-10x+20$$

Add -28 and 20 to get -8.

$$5x+10=7x^{2}-8-10x$$

Subtract 7x^{2} from both sides.

$$5x+10-7x^{2}=-8-10x$$

Subtract -8 from both sides.

$$5x+10-7x^{2}-\left(-8\right)=-10x$$

The opposite of -8 is 8.

$$5x+10-7x^{2}+8=-10x$$

Add 10x to both sides.

$$5x+10-7x^{2}+8+10x=0$$

Add 10 and 8 to get 18.

$$5x+18-7x^{2}+10x=0$$

Combine 5x and 10x to get 15x.

$$15x+18-7x^{2}=0$$

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.

$$-7x^{2}+15x+18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -7 for a, 15 for b, and 18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-15±\sqrt{15^{2}-4\left(-7\right)\times 18}}{2\left(-7\right)}$$

Square 15.

$$x=\frac{-15±\sqrt{225-4\left(-7\right)\times 18}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$x=\frac{-15±\sqrt{225+28\times 18}}{2\left(-7\right)}$$

Multiply 28 times 18.

$$x=\frac{-15±\sqrt{225+504}}{2\left(-7\right)}$$

Add 225 to 504.

$$x=\frac{-15±\sqrt{729}}{2\left(-7\right)}$$

Take the square root of 729.

$$x=\frac{-15±27}{2\left(-7\right)}$$

Multiply 2 times -7.

$$x=\frac{-15±27}{-14}$$

Now solve the equation x=\frac{-15±27}{-14} when ± is plus. Add -15 to 27.

$$x=\frac{12}{-14}$$

Reduce the fraction \frac{12}{-14} to lowest terms by extracting and canceling out 2.

$$x=-\frac{6}{7}$$

Now solve the equation x=\frac{-15±27}{-14} when ± is minus. Subtract 27 from -15.

$$x=-\frac{42}{-14}$$

Divide -42 by -14.

$$x=3$$

The equation is now solved.

$$x=-\frac{6}{7} x=3$$

Steps for Completing the Square

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

$$\left(x+2\right)\times 5=\left(x-2\right)\left(x+2\right)\times 7-\left(x-2\right)\times 10$$

Use the distributive property to multiply x+2 by 5.

$$5x+10=\left(x-2\right)\left(x+2\right)\times 7-\left(x-2\right)\times 10$$

Use the distributive property to multiply x-2 by x+2 and combine like terms.

$$5x+10=\left(x^{2}-4\right)\times 7-\left(x-2\right)\times 10$$

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

$$5x+10=7x^{2}-28-\left(x-2\right)\times 10$$

Use the distributive property to multiply x-2 by 10.

$$5x+10=7x^{2}-28-\left(10x-20\right)$$

To find the opposite of 10x-20, find the opposite of each term.

$$5x+10=7x^{2}-28-10x+20$$

Add -28 and 20 to get -8.

$$5x+10=7x^{2}-8-10x$$

Subtract 7x^{2} from both sides.

$$5x+10-7x^{2}=-8-10x$$

Add 10x to both sides.

$$5x+10-7x^{2}+10x=-8$$

Combine 5x and 10x to get 15x.

$$15x+10-7x^{2}=-8$$

Subtract 10 from both sides.

$$15x-7x^{2}=-8-10$$

Subtract 10 from -8 to get -18.

$$15x-7x^{2}=-18$$

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.

$$-7x^{2}+15x=-18$$

Divide both sides by -7.

$$\frac{-7x^{2}+15x}{-7}=-\frac{18}{-7}$$

Dividing by -7 undoes the multiplication by -7.

$$x^{2}+\frac{15}{-7}x=-\frac{18}{-7}$$

Divide 15 by -7.

$$x^{2}-\frac{15}{7}x=-\frac{18}{-7}$$

Divide -18 by -7.

$$x^{2}-\frac{15}{7}x=\frac{18}{7}$$

Divide -\frac{15}{7}, the coefficient of the x term, by 2 to get -\frac{15}{14}. Then add the square of -\frac{15}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{15}{7}x+\left(-\frac{15}{14}\right)^{2}=\frac{18}{7}+\left(-\frac{15}{14}\right)^{2}$$

Square -\frac{15}{14} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{15}{7}x+\frac{225}{196}=\frac{18}{7}+\frac{225}{196}$$

Add \frac{18}{7} to \frac{225}{196} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{15}{7}x+\frac{225}{196}=\frac{729}{196}$$

Factor x^{2}-\frac{15}{7}x+\frac{225}{196}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{15}{14}\right)^{2}=\frac{729}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{15}{14}\right)^{2}}=\sqrt{\frac{729}{196}}$$

Simplify.

$$x-\frac{15}{14}=\frac{27}{14} x-\frac{15}{14}=-\frac{27}{14}$$

Add \frac{15}{14} to both sides of the equation.

$$x=3 x=-\frac{6}{7}$$