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

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

Solve for x

$$x=\frac{3-\sqrt{13}}{4}\approx -0.151387819$$

$$x = \frac{\sqrt{13} + 3}{4} \approx 1.651387819$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -\frac{1}{2} since division by zero is not defined. Multiply both sides of the equation by 2x+1.

$$2x\left(2x+1\right)+3=4\left(2x+1\right)$$

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

$$4x^{2}+2x+3=4\left(2x+1\right)$$

Use the distributive property to multiply 4 by 2x+1.

$$4x^{2}+2x+3=8x+4$$

Subtract 8x from both sides.

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

Combine 2x and -8x to get -6x.

$$4x^{2}-6x+3=4$$

Subtract 4 from both sides.

$$4x^{2}-6x+3-4=0$$

Subtract 4 from 3 to get -1.

$$4x^{2}-6x-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -6 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 4\left(-1\right)}}{2\times 4}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 4\left(-1\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-6\right)±\sqrt{36-16\left(-1\right)}}{2\times 4}$$

Multiply -16 times -1.

$$x=\frac{-\left(-6\right)±\sqrt{36+16}}{2\times 4}$$

Add 36 to 16.

$$x=\frac{-\left(-6\right)±\sqrt{52}}{2\times 4}$$

Take the square root of 52.

$$x=\frac{-\left(-6\right)±2\sqrt{13}}{2\times 4}$$

The opposite of -6 is 6.

$$x=\frac{6±2\sqrt{13}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{6±2\sqrt{13}}{8}$$

Now solve the equation x=\frac{6±2\sqrt{13}}{8} when ± is plus. Add 6 to 2\sqrt{13}.

$$x=\frac{2\sqrt{13}+6}{8}$$

Divide 6+2\sqrt{13} by 8.

$$x=\frac{\sqrt{13}+3}{4}$$

Now solve the equation x=\frac{6±2\sqrt{13}}{8} when ± is minus. Subtract 2\sqrt{13} from 6.

$$x=\frac{6-2\sqrt{13}}{8}$$

Divide 6-2\sqrt{13} by 8.

$$x=\frac{3-\sqrt{13}}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{13}+3}{4} x=\frac{3-\sqrt{13}}{4}$$

Steps for Completing the Square

Variable x cannot be equal to -\frac{1}{2} since division by zero is not defined. Multiply both sides of the equation by 2x+1.

$$2x\left(2x+1\right)+3=4\left(2x+1\right)$$

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

$$4x^{2}+2x+3=4\left(2x+1\right)$$

Use the distributive property to multiply 4 by 2x+1.

$$4x^{2}+2x+3=8x+4$$

Subtract 8x from both sides.

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

Combine 2x and -8x to get -6x.

$$4x^{2}-6x+3=4$$

Subtract 3 from both sides.

$$4x^{2}-6x=4-3$$

Subtract 3 from 4 to get 1.

$$4x^{2}-6x=1$$

Divide both sides by 4.

$$\frac{4x^{2}-6x}{4}=\frac{1}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\left(-\frac{6}{4}\right)x=\frac{1}{4}$$

Reduce the fraction \frac{-6}{4} to lowest terms by extracting and canceling out 2.

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

Divide -\frac{3}{2}, the coefficient of the x term, by 2 to get -\frac{3}{4}. Then add the square of -\frac{3}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{3}{2}x+\left(-\frac{3}{4}\right)^{2}=\frac{1}{4}+\left(-\frac{3}{4}\right)^{2}$$

Square -\frac{3}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{1}{4}+\frac{9}{16}$$

Add \frac{1}{4} to \frac{9}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{13}{16}$$

Factor x^{2}-\frac{3}{2}x+\frac{9}{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}.

$$\left(x-\frac{3}{4}\right)^{2}=\frac{13}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{4}\right)^{2}}=\sqrt{\frac{13}{16}}$$

Simplify.

$$x-\frac{3}{4}=\frac{\sqrt{13}}{4} x-\frac{3}{4}=-\frac{\sqrt{13}}{4}$$

Add \frac{3}{4} to both sides of the equation.

$$x=\frac{\sqrt{13}+3}{4} x=\frac{3-\sqrt{13}}{4}$$