Solve 1/(x+2)=4x+5

$$\frac { 1 } { ( x + 2 ) } = 4 x + 5$$

Solve for x

$$x = -\frac{9}{4} = -2\frac{1}{4} = -2.25$$

$$x=-1$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -2 since division by zero is not defined. Multiply both sides of the equation by x+2.

$$1=4x\left(x+2\right)+\left(x+2\right)\times 5$$

Use the distributive property to multiply 4x by x+2.

$$1=4x^{2}+8x+\left(x+2\right)\times 5$$

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

$$1=4x^{2}+8x+5x+10$$

Combine 8x and 5x to get 13x.

$$1=4x^{2}+13x+10$$

Swap sides so that all variable terms are on the left hand side.

$$4x^{2}+13x+10=1$$

Subtract 1 from both sides.

$$4x^{2}+13x+10-1=0$$

Subtract 1 from 10 to get 9.

$$4x^{2}+13x+9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 13 for b, and 9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-13±\sqrt{13^{2}-4\times 4\times 9}}{2\times 4}$$

Square 13.

$$x=\frac{-13±\sqrt{169-4\times 4\times 9}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-13±\sqrt{169-16\times 9}}{2\times 4}$$

Multiply -16 times 9.

$$x=\frac{-13±\sqrt{169-144}}{2\times 4}$$

Add 169 to -144.

$$x=\frac{-13±\sqrt{25}}{2\times 4}$$

Take the square root of 25.

$$x=\frac{-13±5}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-13±5}{8}$$

Now solve the equation x=\frac{-13±5}{8} when ± is plus. Add -13 to 5.

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

Divide -8 by 8.

$$x=-1$$

Now solve the equation x=\frac{-13±5}{8} when ± is minus. Subtract 5 from -13.

$$x=-\frac{18}{8}$$

Reduce the fraction \frac{-18}{8} to lowest terms by extracting and canceling out 2.

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

The equation is now solved.

$$x=-1 x=-\frac{9}{4}$$

Steps for Completing the Square

Variable x cannot be equal to -2 since division by zero is not defined. Multiply both sides of the equation by x+2.

$$1=4x\left(x+2\right)+\left(x+2\right)\times 5$$

Use the distributive property to multiply 4x by x+2.

$$1=4x^{2}+8x+\left(x+2\right)\times 5$$

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

$$1=4x^{2}+8x+5x+10$$

Combine 8x and 5x to get 13x.

$$1=4x^{2}+13x+10$$

Swap sides so that all variable terms are on the left hand side.

$$4x^{2}+13x+10=1$$

Subtract 10 from both sides.

$$4x^{2}+13x=1-10$$

Subtract 10 from 1 to get -9.

$$4x^{2}+13x=-9$$

Divide both sides by 4.

$$\frac{4x^{2}+13x}{4}=-\frac{9}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{13}{4}x=-\frac{9}{4}$$

Divide \frac{13}{4}, the coefficient of the x term, by 2 to get \frac{13}{8}. Then add the square of \frac{13}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{13}{4}x+\left(\frac{13}{8}\right)^{2}=-\frac{9}{4}+\left(\frac{13}{8}\right)^{2}$$

Square \frac{13}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{13}{4}x+\frac{169}{64}=-\frac{9}{4}+\frac{169}{64}$$

Add -\frac{9}{4} to \frac{169}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{13}{4}x+\frac{169}{64}=\frac{25}{64}$$

Factor x^{2}+\frac{13}{4}x+\frac{169}{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}.

$$\left(x+\frac{13}{8}\right)^{2}=\frac{25}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{13}{8}\right)^{2}}=\sqrt{\frac{25}{64}}$$

Simplify.

$$x+\frac{13}{8}=\frac{5}{8} x+\frac{13}{8}=-\frac{5}{8}$$

Subtract \frac{13}{8} from both sides of the equation.

$$x=-1 x=-\frac{9}{4}$$