Solve x+1/2x-3+3x+1/2x+3=4x+6/4x^2-9

$$\frac { x + 1 } { 2 x - 3 } + \frac { 3 x + 1 } { 2 x + 3 } = \frac { 4 x + 6 } { 4 x ^ { 2 } - 9 }$$

Solve for x

$$x = \frac{\sqrt{57} + 3}{8} \approx 1.318729304$$

$$x=\frac{3-\sqrt{57}}{8}\approx -0.568729304$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -\frac{3}{2},\frac{3}{2} since division by zero is not defined. Multiply both sides of the equation by \left(2x-3\right)\left(2x+3\right), the least common multiple of 2x-3,2x+3,4x^{2}-9.

$$\left(2x+3\right)\left(x+1\right)+\left(2x-3\right)\left(3x+1\right)=4x+6$$

Use the distributive property to multiply 2x+3 by x+1 and combine like terms.

$$2x^{2}+5x+3+\left(2x-3\right)\left(3x+1\right)=4x+6$$

Use the distributive property to multiply 2x-3 by 3x+1 and combine like terms.

$$2x^{2}+5x+3+6x^{2}-7x-3=4x+6$$

Combine 2x^{2} and 6x^{2} to get 8x^{2}.

$$8x^{2}+5x+3-7x-3=4x+6$$

Combine 5x and -7x to get -2x.

$$8x^{2}-2x+3-3=4x+6$$

Subtract 3 from 3 to get 0.

$$8x^{2}-2x=4x+6$$

Subtract 4x from both sides.

$$8x^{2}-2x-4x=6$$

Combine -2x and -4x to get -6x.

$$8x^{2}-6x=6$$

Subtract 6 from both sides.

$$8x^{2}-6x-6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, -6 for b, and -6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 8\left(-6\right)}}{2\times 8}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 8\left(-6\right)}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-\left(-6\right)±\sqrt{36-32\left(-6\right)}}{2\times 8}$$

Multiply -32 times -6.

$$x=\frac{-\left(-6\right)±\sqrt{36+192}}{2\times 8}$$

Add 36 to 192.

$$x=\frac{-\left(-6\right)±\sqrt{228}}{2\times 8}$$

Take the square root of 228.

$$x=\frac{-\left(-6\right)±2\sqrt{57}}{2\times 8}$$

The opposite of -6 is 6.

$$x=\frac{6±2\sqrt{57}}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{6±2\sqrt{57}}{16}$$

Now solve the equation x=\frac{6±2\sqrt{57}}{16} when ± is plus. Add 6 to 2\sqrt{57}.

$$x=\frac{2\sqrt{57}+6}{16}$$

Divide 6+2\sqrt{57} by 16.

$$x=\frac{\sqrt{57}+3}{8}$$

Now solve the equation x=\frac{6±2\sqrt{57}}{16} when ± is minus. Subtract 2\sqrt{57} from 6.

$$x=\frac{6-2\sqrt{57}}{16}$$

Divide 6-2\sqrt{57} by 16.

$$x=\frac{3-\sqrt{57}}{8}$$

The equation is now solved.

$$x=\frac{\sqrt{57}+3}{8} x=\frac{3-\sqrt{57}}{8}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -\frac{3}{2},\frac{3}{2} since division by zero is not defined. Multiply both sides of the equation by \left(2x-3\right)\left(2x+3\right), the least common multiple of 2x-3,2x+3,4x^{2}-9.

$$\left(2x+3\right)\left(x+1\right)+\left(2x-3\right)\left(3x+1\right)=4x+6$$

Use the distributive property to multiply 2x+3 by x+1 and combine like terms.

$$2x^{2}+5x+3+\left(2x-3\right)\left(3x+1\right)=4x+6$$

Use the distributive property to multiply 2x-3 by 3x+1 and combine like terms.

$$2x^{2}+5x+3+6x^{2}-7x-3=4x+6$$

Combine 2x^{2} and 6x^{2} to get 8x^{2}.

$$8x^{2}+5x+3-7x-3=4x+6$$

Combine 5x and -7x to get -2x.

$$8x^{2}-2x+3-3=4x+6$$

Subtract 3 from 3 to get 0.

$$8x^{2}-2x=4x+6$$

Subtract 4x from both sides.

$$8x^{2}-2x-4x=6$$

Combine -2x and -4x to get -6x.

$$8x^{2}-6x=6$$

Divide both sides by 8.

$$\frac{8x^{2}-6x}{8}=\frac{6}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$x^{2}+\left(-\frac{6}{8}\right)x=\frac{6}{8}$$

Reduce the fraction \frac{-6}{8} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{3}{4}x=\frac{6}{8}$$

Reduce the fraction \frac{6}{8} to lowest terms by extracting and canceling out 2.

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

Divide -\frac{3}{4}, the coefficient of the x term, by 2 to get -\frac{3}{8}. Then add the square of -\frac{3}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{3}{4}x+\left(-\frac{3}{8}\right)^{2}=\frac{3}{4}+\left(-\frac{3}{8}\right)^{2}$$

Square -\frac{3}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{3}{4}x+\frac{9}{64}=\frac{3}{4}+\frac{9}{64}$$

Add \frac{3}{4} to \frac{9}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{3}{4}x+\frac{9}{64}=\frac{57}{64}$$

Factor x^{2}-\frac{3}{4}x+\frac{9}{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}.

$$\left(x-\frac{3}{8}\right)^{2}=\frac{57}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{8}\right)^{2}}=\sqrt{\frac{57}{64}}$$

Simplify.

$$x-\frac{3}{8}=\frac{\sqrt{57}}{8} x-\frac{3}{8}=-\frac{\sqrt{57}}{8}$$

Add \frac{3}{8} to both sides of the equation.

$$x=\frac{\sqrt{57}+3}{8} x=\frac{3-\sqrt{57}}{8}$$