Solve 1/x+2=5x-4/3

$$\frac { 1 } { x + 2 } = \frac { 5 x - 4 } { 3 }$$

Solve for x

$$x = -\frac{11}{5} = -2\frac{1}{5} = -2.2$$

$$x=1$$

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 3\left(x+2\right), the least common multiple of x+2,3.

$$3=\left(x+2\right)\left(5x-4\right)$$

Use the distributive property to multiply x+2 by 5x-4 and combine like terms.

$$3=5x^{2}+6x-8$$

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

$$5x^{2}+6x-8=3$$

Subtract 3 from both sides.

$$5x^{2}+6x-8-3=0$$

Subtract 3 from -8 to get -11.

$$5x^{2}+6x-11=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 6 for b, and -11 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\times 5\left(-11\right)}}{2\times 5}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\times 5\left(-11\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-6±\sqrt{36-20\left(-11\right)}}{2\times 5}$$

Multiply -20 times -11.

$$x=\frac{-6±\sqrt{36+220}}{2\times 5}$$

Add 36 to 220.

$$x=\frac{-6±\sqrt{256}}{2\times 5}$$

Take the square root of 256.

$$x=\frac{-6±16}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{-6±16}{10}$$

Now solve the equation x=\frac{-6±16}{10} when ± is plus. Add -6 to 16.

$$x=\frac{10}{10}$$

Divide 10 by 10.

$$x=1$$

Now solve the equation x=\frac{-6±16}{10} when ± is minus. Subtract 16 from -6.

$$x=-\frac{22}{10}$$

Reduce the fraction \frac{-22}{10} to lowest terms by extracting and canceling out 2.

$$x=-\frac{11}{5}$$

The equation is now solved.

$$x=1 x=-\frac{11}{5}$$

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 3\left(x+2\right), the least common multiple of x+2,3.

$$3=\left(x+2\right)\left(5x-4\right)$$

Use the distributive property to multiply x+2 by 5x-4 and combine like terms.

$$3=5x^{2}+6x-8$$

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

$$5x^{2}+6x-8=3$$

Add 8 to both sides.

$$5x^{2}+6x=3+8$$

Add 3 and 8 to get 11.

$$5x^{2}+6x=11$$

Divide both sides by 5.

$$\frac{5x^{2}+6x}{5}=\frac{11}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}+\frac{6}{5}x=\frac{11}{5}$$

Divide \frac{6}{5}, the coefficient of the x term, by 2 to get \frac{3}{5}. Then add the square of \frac{3}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{6}{5}x+\left(\frac{3}{5}\right)^{2}=\frac{11}{5}+\left(\frac{3}{5}\right)^{2}$$

Square \frac{3}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{6}{5}x+\frac{9}{25}=\frac{11}{5}+\frac{9}{25}$$

Add \frac{11}{5} to \frac{9}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{6}{5}x+\frac{9}{25}=\frac{64}{25}$$

Factor x^{2}+\frac{6}{5}x+\frac{9}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{3}{5}\right)^{2}=\frac{64}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{5}\right)^{2}}=\sqrt{\frac{64}{25}}$$

Simplify.

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

Subtract \frac{3}{5} from both sides of the equation.

$$x=1 x=-\frac{11}{5}$$