Solve 2x+3/2x-3+2x-3/2x+3=17/4

$$\frac { 2 x + 3 } { 2 x - 3 } + \frac { 2 x - 3 } { 2 x + 3 } = \frac { 17 } { 4 }$$

Solve for x

$$x = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

$$x = -\frac{5}{2} = -2\frac{1}{2} = -2.5$$

Steps by Finding Square Root

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 4\left(2x-3\right)\left(2x+3\right), the least common multiple of 2x-3,2x+3,4.

$$\left(8x+12\right)\left(2x+3\right)+\left(8x-12\right)\left(2x-3\right)=17\left(2x-3\right)\left(2x+3\right)$$

Use the distributive property to multiply 8x+12 by 2x+3 and combine like terms.

$$16x^{2}+48x+36+\left(8x-12\right)\left(2x-3\right)=17\left(2x-3\right)\left(2x+3\right)$$

Use the distributive property to multiply 8x-12 by 2x-3 and combine like terms.

$$16x^{2}+48x+36+16x^{2}-48x+36=17\left(2x-3\right)\left(2x+3\right)$$

Combine 16x^{2} and 16x^{2} to get 32x^{2}.

$$32x^{2}+48x+36-48x+36=17\left(2x-3\right)\left(2x+3\right)$$

Combine 48x and -48x to get 0.

$$32x^{2}+36+36=17\left(2x-3\right)\left(2x+3\right)$$

Add 36 and 36 to get 72.

$$32x^{2}+72=17\left(2x-3\right)\left(2x+3\right)$$

Use the distributive property to multiply 17 by 2x-3.

$$32x^{2}+72=\left(34x-51\right)\left(2x+3\right)$$

Use the distributive property to multiply 34x-51 by 2x+3 and combine like terms.

$$32x^{2}+72=68x^{2}-153$$

Subtract 68x^{2} from both sides.

$$32x^{2}+72-68x^{2}=-153$$

Combine 32x^{2} and -68x^{2} to get -36x^{2}.

$$-36x^{2}+72=-153$$

Subtract 72 from both sides.

$$-36x^{2}=-153-72$$

Subtract 72 from -153 to get -225.

$$-36x^{2}=-225$$

Divide both sides by -36.

$$x^{2}=\frac{-225}{-36}$$

Reduce the fraction \frac{-225}{-36} to lowest terms by extracting and canceling out -9.

$$x^{2}=\frac{25}{4}$$

Take the square root of both sides of the equation.

$$x=\frac{5}{2} x=-\frac{5}{2}$$

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 4\left(2x-3\right)\left(2x+3\right), the least common multiple of 2x-3,2x+3,4.

$$\left(8x+12\right)\left(2x+3\right)+\left(8x-12\right)\left(2x-3\right)=17\left(2x-3\right)\left(2x+3\right)$$

Use the distributive property to multiply 8x+12 by 2x+3 and combine like terms.

$$16x^{2}+48x+36+\left(8x-12\right)\left(2x-3\right)=17\left(2x-3\right)\left(2x+3\right)$$

Use the distributive property to multiply 8x-12 by 2x-3 and combine like terms.

$$16x^{2}+48x+36+16x^{2}-48x+36=17\left(2x-3\right)\left(2x+3\right)$$

Combine 16x^{2} and 16x^{2} to get 32x^{2}.

$$32x^{2}+48x+36-48x+36=17\left(2x-3\right)\left(2x+3\right)$$

Combine 48x and -48x to get 0.

$$32x^{2}+36+36=17\left(2x-3\right)\left(2x+3\right)$$

Add 36 and 36 to get 72.

$$32x^{2}+72=17\left(2x-3\right)\left(2x+3\right)$$

Use the distributive property to multiply 17 by 2x-3.

$$32x^{2}+72=\left(34x-51\right)\left(2x+3\right)$$

Use the distributive property to multiply 34x-51 by 2x+3 and combine like terms.

$$32x^{2}+72=68x^{2}-153$$

Subtract 68x^{2} from both sides.

$$32x^{2}+72-68x^{2}=-153$$

Combine 32x^{2} and -68x^{2} to get -36x^{2}.

$$-36x^{2}+72=-153$$

Add 153 to both sides.

$$-36x^{2}+72+153=0$$

Add 72 and 153 to get 225.

$$-36x^{2}+225=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -36 for a, 0 for b, and 225 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-36\right)\times 225}}{2\left(-36\right)}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-36\right)\times 225}}{2\left(-36\right)}$$

Multiply -4 times -36.

$$x=\frac{0±\sqrt{144\times 225}}{2\left(-36\right)}$$

Multiply 144 times 225.

$$x=\frac{0±\sqrt{32400}}{2\left(-36\right)}$$

Take the square root of 32400.

$$x=\frac{0±180}{2\left(-36\right)}$$

Multiply 2 times -36.

$$x=\frac{0±180}{-72}$$

Now solve the equation x=\frac{0±180}{-72} when ± is plus. Reduce the fraction \frac{180}{-72} to lowest terms by extracting and canceling out 36.

$$x=-\frac{5}{2}$$

Now solve the equation x=\frac{0±180}{-72} when ± is minus. Reduce the fraction \frac{-180}{-72} to lowest terms by extracting and canceling out 36.

$$x=\frac{5}{2}$$

The equation is now solved.

$$x=-\frac{5}{2} x=\frac{5}{2}$$