Solve x-3/2+4/2x+3=0

$$x - \frac { 3 } { 2 } + \frac { 4 } { 2 x + 3 } = 0$$

Solve for x

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

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

Steps Using Difference of Squares

Variable x cannot be equal to -\frac{3}{2} since division by zero is not defined. Multiply both sides of the equation by 2\left(2x+3\right), the least common multiple of 2,2x+3.

$$2\left(2x+3\right)x+2\left(2x+3\right)\left(-\frac{3}{2}\right)+2\times 4=0$$

Use the distributive property to multiply 2 by 2x+3.

$$\left(4x+6\right)x+2\left(2x+3\right)\left(-\frac{3}{2}\right)+2\times 4=0$$

Use the distributive property to multiply 4x+6 by x.

$$4x^{2}+6x+2\left(2x+3\right)\left(-\frac{3}{2}\right)+2\times 4=0$$

Multiply 2 and -\frac{3}{2} to get -3.

$$4x^{2}+6x-3\left(2x+3\right)+2\times 4=0$$

Use the distributive property to multiply -3 by 2x+3.

$$4x^{2}+6x-6x-9+2\times 4=0$$

Combine 6x and -6x to get 0.

$$4x^{2}-9+2\times 4=0$$

Multiply 2 and 4 to get 8.

$$4x^{2}-9+8=0$$

Add -9 and 8 to get -1.

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

Consider 4x^{2}-1. Rewrite 4x^{2}-1 as \left(2x\right)^{2}-1^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(2x-1\right)\left(2x+1\right)=0$$

To find equation solutions, solve 2x-1=0 and 2x+1=0.

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

Steps by Finding Square Root

Variable x cannot be equal to -\frac{3}{2} since division by zero is not defined. Multiply both sides of the equation by 2\left(2x+3\right), the least common multiple of 2,2x+3.

$$2\left(2x+3\right)x+2\left(2x+3\right)\left(-\frac{3}{2}\right)+2\times 4=0$$

Use the distributive property to multiply 2 by 2x+3.

$$\left(4x+6\right)x+2\left(2x+3\right)\left(-\frac{3}{2}\right)+2\times 4=0$$

Use the distributive property to multiply 4x+6 by x.

$$4x^{2}+6x+2\left(2x+3\right)\left(-\frac{3}{2}\right)+2\times 4=0$$

Multiply 2 and -\frac{3}{2} to get -3.

$$4x^{2}+6x-3\left(2x+3\right)+2\times 4=0$$

Use the distributive property to multiply -3 by 2x+3.

$$4x^{2}+6x-6x-9+2\times 4=0$$

Combine 6x and -6x to get 0.

$$4x^{2}-9+2\times 4=0$$

Multiply 2 and 4 to get 8.

$$4x^{2}-9+8=0$$

Add -9 and 8 to get -1.

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

Add 1 to both sides. Anything plus zero gives itself.

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

Divide both sides by 4.

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

Take the square root of both sides of the equation.

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

Steps Using the Quadratic Formula

Variable x cannot be equal to -\frac{3}{2} since division by zero is not defined. Multiply both sides of the equation by 2\left(2x+3\right), the least common multiple of 2,2x+3.

$$2\left(2x+3\right)x+2\left(2x+3\right)\left(-\frac{3}{2}\right)+2\times 4=0$$

Use the distributive property to multiply 2 by 2x+3.

$$\left(4x+6\right)x+2\left(2x+3\right)\left(-\frac{3}{2}\right)+2\times 4=0$$

Use the distributive property to multiply 4x+6 by x.

$$4x^{2}+6x+2\left(2x+3\right)\left(-\frac{3}{2}\right)+2\times 4=0$$

Multiply 2 and -\frac{3}{2} to get -3.

$$4x^{2}+6x-3\left(2x+3\right)+2\times 4=0$$

Use the distributive property to multiply -3 by 2x+3.

$$4x^{2}+6x-6x-9+2\times 4=0$$

Combine 6x and -6x to get 0.

$$4x^{2}-9+2\times 4=0$$

Multiply 2 and 4 to get 8.

$$4x^{2}-9+8=0$$

Add -9 and 8 to get -1.

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

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

$$x=\frac{0±\sqrt{0^{2}-4\times 4\left(-1\right)}}{2\times 4}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 4\left(-1\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{0±\sqrt{-16\left(-1\right)}}{2\times 4}$$

Multiply -16 times -1.

$$x=\frac{0±\sqrt{16}}{2\times 4}$$

Take the square root of 16.

$$x=\frac{0±4}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{0±4}{8}$$

Now solve the equation x=\frac{0±4}{8} when ± is plus. Reduce the fraction \frac{4}{8} to lowest terms by extracting and canceling out 4.

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

Now solve the equation x=\frac{0±4}{8} when ± is minus. Reduce the fraction \frac{-4}{8} to lowest terms by extracting and canceling out 4.

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

The equation is now solved.

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