Solve x+3/x-3+x-3/x+3=21/2

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

Solve for x

$$x=9$$

$$x=-9$$

Steps by Finding Square Root

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

$$\left(2x+6\right)\left(x+3\right)+\left(2x-6\right)\left(x-3\right)=\left(x^{2}-9\right)\left(2\times 2+1\right)$$

Use the distributive property to multiply 2x+6 by x+3 and combine like terms.

$$2x^{2}+12x+18+\left(2x-6\right)\left(x-3\right)=\left(x^{2}-9\right)\left(2\times 2+1\right)$$

Use the distributive property to multiply 2x-6 by x-3 and combine like terms.

$$2x^{2}+12x+18+2x^{2}-12x+18=\left(x^{2}-9\right)\left(2\times 2+1\right)$$

Combine 2x^{2} and 2x^{2} to get 4x^{2}.

$$4x^{2}+12x+18-12x+18=\left(x^{2}-9\right)\left(2\times 2+1\right)$$

Combine 12x and -12x to get 0.

$$4x^{2}+18+18=\left(x^{2}-9\right)\left(2\times 2+1\right)$$

Add 18 and 18 to get 36.

$$4x^{2}+36=\left(x^{2}-9\right)\left(2\times 2+1\right)$$

Multiply 2 and 2 to get 4.

$$4x^{2}+36=\left(x^{2}-9\right)\left(4+1\right)$$

Add 4 and 1 to get 5.

$$4x^{2}+36=\left(x^{2}-9\right)\times 5$$

Use the distributive property to multiply x^{2}-9 by 5.

$$4x^{2}+36=5x^{2}-45$$

Subtract 5x^{2} from both sides.

$$4x^{2}+36-5x^{2}=-45$$

Combine 4x^{2} and -5x^{2} to get -x^{2}.

$$-x^{2}+36=-45$$

Subtract 36 from both sides.

$$-x^{2}=-45-36$$

Subtract 36 from -45 to get -81.

$$-x^{2}=-81$$

Divide both sides by -1.

$$x^{2}=\frac{-81}{-1}$$

Fraction \frac{-81}{-1} can be simplified to 81 by removing the negative sign from both the numerator and the denominator.

$$x^{2}=81$$

Take the square root of both sides of the equation.

$$x=9 x=-9$$

Steps Using the Quadratic Formula

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

$$\left(2x+6\right)\left(x+3\right)+\left(2x-6\right)\left(x-3\right)=\left(x^{2}-9\right)\left(2\times 2+1\right)$$

Use the distributive property to multiply 2x+6 by x+3 and combine like terms.

$$2x^{2}+12x+18+\left(2x-6\right)\left(x-3\right)=\left(x^{2}-9\right)\left(2\times 2+1\right)$$

Use the distributive property to multiply 2x-6 by x-3 and combine like terms.

$$2x^{2}+12x+18+2x^{2}-12x+18=\left(x^{2}-9\right)\left(2\times 2+1\right)$$

Combine 2x^{2} and 2x^{2} to get 4x^{2}.

$$4x^{2}+12x+18-12x+18=\left(x^{2}-9\right)\left(2\times 2+1\right)$$

Combine 12x and -12x to get 0.

$$4x^{2}+18+18=\left(x^{2}-9\right)\left(2\times 2+1\right)$$

Add 18 and 18 to get 36.

$$4x^{2}+36=\left(x^{2}-9\right)\left(2\times 2+1\right)$$

Multiply 2 and 2 to get 4.

$$4x^{2}+36=\left(x^{2}-9\right)\left(4+1\right)$$

Add 4 and 1 to get 5.

$$4x^{2}+36=\left(x^{2}-9\right)\times 5$$

Use the distributive property to multiply x^{2}-9 by 5.

$$4x^{2}+36=5x^{2}-45$$

Subtract 5x^{2} from both sides.

$$4x^{2}+36-5x^{2}=-45$$

Combine 4x^{2} and -5x^{2} to get -x^{2}.

$$-x^{2}+36=-45$$

Add 45 to both sides.

$$-x^{2}+36+45=0$$

Add 36 and 45 to get 81.

$$-x^{2}+81=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 0 for b, and 81 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-1\right)\times 81}}{2\left(-1\right)}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-1\right)\times 81}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{0±\sqrt{4\times 81}}{2\left(-1\right)}$$

Multiply 4 times 81.

$$x=\frac{0±\sqrt{324}}{2\left(-1\right)}$$

Take the square root of 324.

$$x=\frac{0±18}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{0±18}{-2}$$

Now solve the equation x=\frac{0±18}{-2} when ± is plus. Divide 18 by -2.

$$x=-9$$

Now solve the equation x=\frac{0±18}{-2} when ± is minus. Divide -18 by -2.

$$x=9$$

The equation is now solved.

$$x=-9 x=9$$