Solve 2x-9/x+1=-5

$$2 x - \frac { 9 } { x + 1 } = - 5$$

Solve for x

$$x=-4$$

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

Steps Using the Quadratic Formula

Variable x cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by x+1.

$$2x\left(x+1\right)-9=-5\left(x+1\right)$$

Use the distributive property to multiply 2x by x+1.

$$2x^{2}+2x-9=-5\left(x+1\right)$$

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

$$2x^{2}+2x-9=-5x-5$$

Add 5x to both sides.

$$2x^{2}+2x-9+5x=-5$$

Combine 2x and 5x to get 7x.

$$2x^{2}+7x-9=-5$$

Add 5 to both sides.

$$2x^{2}+7x-9+5=0$$

Add -9 and 5 to get -4.

$$2x^{2}+7x-4=0$$

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

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

Square 7.

$$x=\frac{-7±\sqrt{49-4\times 2\left(-4\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-7±\sqrt{49-8\left(-4\right)}}{2\times 2}$$

Multiply -8 times -4.

$$x=\frac{-7±\sqrt{49+32}}{2\times 2}$$

Add 49 to 32.

$$x=\frac{-7±\sqrt{81}}{2\times 2}$$

Take the square root of 81.

$$x=\frac{-7±9}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-7±9}{4}$$

Now solve the equation x=\frac{-7±9}{4} when ± is plus. Add -7 to 9.

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

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

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

Now solve the equation x=\frac{-7±9}{4} when ± is minus. Subtract 9 from -7.

$$x=-\frac{16}{4}$$

Divide -16 by 4.

$$x=-4$$

The equation is now solved.

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

Steps for Completing the Square

Variable x cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by x+1.

$$2x\left(x+1\right)-9=-5\left(x+1\right)$$

Use the distributive property to multiply 2x by x+1.

$$2x^{2}+2x-9=-5\left(x+1\right)$$

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

$$2x^{2}+2x-9=-5x-5$$

Add 5x to both sides.

$$2x^{2}+2x-9+5x=-5$$

Combine 2x and 5x to get 7x.

$$2x^{2}+7x-9=-5$$

Add 9 to both sides.

$$2x^{2}+7x=-5+9$$

Add -5 and 9 to get 4.

$$2x^{2}+7x=4$$

Divide both sides by 2.

$$\frac{2x^{2}+7x}{2}=\frac{4}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{7}{2}x=\frac{4}{2}$$

Divide 4 by 2.

$$x^{2}+\frac{7}{2}x=2$$

Divide \frac{7}{2}, the coefficient of the x term, by 2 to get \frac{7}{4}. Then add the square of \frac{7}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square \frac{7}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{7}{2}x+\frac{49}{16}=2+\frac{49}{16}$$

Add 2 to \frac{49}{16}.

$$x^{2}+\frac{7}{2}x+\frac{49}{16}=\frac{81}{16}$$

Factor x^{2}+\frac{7}{2}x+\frac{49}{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}.

$$\left(x+\frac{7}{4}\right)^{2}=\frac{81}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{4}\right)^{2}}=\sqrt{\frac{81}{16}}$$

Simplify.

$$x+\frac{7}{4}=\frac{9}{4} x+\frac{7}{4}=-\frac{9}{4}$$

Subtract \frac{7}{4} from both sides of the equation.

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