Solve 7/x-1=-8+5/x+2

$$\frac { 7 } { x - 1 } = - 8 + \frac { 5 } { x + 2 }$$

Solve for x

$$x=-\frac{3}{4}=-0.75$$

$$x=-\frac{1}{2}=-0.5$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(x+2\right), the least common multiple of x-1,x+2.

$$\left(x+2\right)\times 7=\left(x-1\right)\left(x+2\right)\left(-8\right)+\left(x-1\right)\times 5$$

Use the distributive property to multiply x+2 by 7.

$$7x+14=\left(x-1\right)\left(x+2\right)\left(-8\right)+\left(x-1\right)\times 5$$

Use the distributive property to multiply x-1 by x+2 and combine like terms.

$$7x+14=\left(x^{2}+x-2\right)\left(-8\right)+\left(x-1\right)\times 5$$

Use the distributive property to multiply x^{2}+x-2 by -8.

$$7x+14=-8x^{2}-8x+16+\left(x-1\right)\times 5$$

Use the distributive property to multiply x-1 by 5.

$$7x+14=-8x^{2}-8x+16+5x-5$$

Combine -8x and 5x to get -3x.

$$7x+14=-8x^{2}-3x+16-5$$

Subtract 5 from 16 to get 11.

$$7x+14=-8x^{2}-3x+11$$

Add 8x^{2} to both sides.

$$7x+14+8x^{2}=-3x+11$$

Add 3x to both sides.

$$7x+14+8x^{2}+3x=11$$

Combine 7x and 3x to get 10x.

$$10x+14+8x^{2}=11$$

Subtract 11 from both sides.

$$10x+14+8x^{2}-11=0$$

Subtract 11 from 14 to get 3.

$$10x+3+8x^{2}=0$$

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.

$$8x^{2}+10x+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, 10 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\times 8\times 3}}{2\times 8}$$

Square 10.

$$x=\frac{-10±\sqrt{100-4\times 8\times 3}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-10±\sqrt{100-32\times 3}}{2\times 8}$$

Multiply -32 times 3.

$$x=\frac{-10±\sqrt{100-96}}{2\times 8}$$

Add 100 to -96.

$$x=\frac{-10±\sqrt{4}}{2\times 8}$$

Take the square root of 4.

$$x=\frac{-10±2}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{-10±2}{16}$$

Now solve the equation x=\frac{-10±2}{16} when ± is plus. Add -10 to 2.

$$x=-\frac{8}{16}$$

Reduce the fraction \frac{-8}{16} to lowest terms by extracting and canceling out 8.

$$x=-\frac{1}{2}$$

Now solve the equation x=\frac{-10±2}{16} when ± is minus. Subtract 2 from -10.

$$x=-\frac{12}{16}$$

Reduce the fraction \frac{-12}{16} to lowest terms by extracting and canceling out 4.

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

The equation is now solved.

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

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(x+2\right), the least common multiple of x-1,x+2.

$$\left(x+2\right)\times 7=\left(x-1\right)\left(x+2\right)\left(-8\right)+\left(x-1\right)\times 5$$

Use the distributive property to multiply x+2 by 7.

$$7x+14=\left(x-1\right)\left(x+2\right)\left(-8\right)+\left(x-1\right)\times 5$$

Use the distributive property to multiply x-1 by x+2 and combine like terms.

$$7x+14=\left(x^{2}+x-2\right)\left(-8\right)+\left(x-1\right)\times 5$$

Use the distributive property to multiply x^{2}+x-2 by -8.

$$7x+14=-8x^{2}-8x+16+\left(x-1\right)\times 5$$

Use the distributive property to multiply x-1 by 5.

$$7x+14=-8x^{2}-8x+16+5x-5$$

Combine -8x and 5x to get -3x.

$$7x+14=-8x^{2}-3x+16-5$$

Subtract 5 from 16 to get 11.

$$7x+14=-8x^{2}-3x+11$$

Add 8x^{2} to both sides.

$$7x+14+8x^{2}=-3x+11$$

Add 3x to both sides.

$$7x+14+8x^{2}+3x=11$$

Combine 7x and 3x to get 10x.

$$10x+14+8x^{2}=11$$

Subtract 14 from both sides.

$$10x+8x^{2}=11-14$$

Subtract 14 from 11 to get -3.

$$10x+8x^{2}=-3$$

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.

$$8x^{2}+10x=-3$$

Divide both sides by 8.

$$\frac{8x^{2}+10x}{8}=-\frac{3}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$x^{2}+\frac{10}{8}x=-\frac{3}{8}$$

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

$$x^{2}+\frac{5}{4}x=-\frac{3}{8}$$

Divide \frac{5}{4}, the coefficient of the x term, by 2 to get \frac{5}{8}. Then add the square of \frac{5}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{5}{4}x+\left(\frac{5}{8}\right)^{2}=-\frac{3}{8}+\left(\frac{5}{8}\right)^{2}$$

Square \frac{5}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{5}{4}x+\frac{25}{64}=-\frac{3}{8}+\frac{25}{64}$$

Add -\frac{3}{8} to \frac{25}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{5}{4}x+\frac{25}{64}=\frac{1}{64}$$

Factor x^{2}+\frac{5}{4}x+\frac{25}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{5}{8}\right)^{2}=\frac{1}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{8}\right)^{2}}=\sqrt{\frac{1}{64}}$$

Simplify.

$$x+\frac{5}{8}=\frac{1}{8} x+\frac{5}{8}=-\frac{1}{8}$$

Subtract \frac{5}{8} from both sides of the equation.

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