Solve 1/x+3-1/x-3+1=0

$$\frac{ 1 }{ x+3 } - \frac{ 1 }{ x-3 } +1 = 0$$

Solve for x

$$x=\sqrt{15}\approx 3.872983346$$

$$x=-\sqrt{15}\approx -3.872983346$$

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

$$x-3-\left(x+3\right)+\left(x-3\right)\left(x+3\right)=0$$

To find the opposite of x+3, find the opposite of each term.

$$x-3-x-3+\left(x-3\right)\left(x+3\right)=0$$

Combine x and -x to get 0.

$$-3-3+\left(x-3\right)\left(x+3\right)=0$$

Subtract 3 from -3 to get -6.

$$-6+\left(x-3\right)\left(x+3\right)=0$$

Consider \left(x-3\right)\left(x+3\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 3.

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

Subtract 9 from -6 to get -15.

$$-15+x^{2}=0$$

Add 15 to both sides. Anything plus zero gives itself.

$$x^{2}=15$$

Take the square root of both sides of the equation.

$$x=\sqrt{15} x=-\sqrt{15}$$

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

$$x-3-\left(x+3\right)+\left(x-3\right)\left(x+3\right)=0$$

To find the opposite of x+3, find the opposite of each term.

$$x-3-x-3+\left(x-3\right)\left(x+3\right)=0$$

Combine x and -x to get 0.

$$-3-3+\left(x-3\right)\left(x+3\right)=0$$

Subtract 3 from -3 to get -6.

$$-6+\left(x-3\right)\left(x+3\right)=0$$

Consider \left(x-3\right)\left(x+3\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 3.

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

Subtract 9 from -6 to get -15.

$$-15+x^{2}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$x^{2}-15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 0 for b, and -15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-15\right)}}{2}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-15\right)}}{2}$$

Multiply -4 times -15.

$$x=\frac{0±\sqrt{60}}{2}$$

Take the square root of 60.

$$x=\frac{0±2\sqrt{15}}{2}$$

Now solve the equation x=\frac{0±2\sqrt{15}}{2} when ± is plus.

$$x=\sqrt{15}$$

Now solve the equation x=\frac{0±2\sqrt{15}}{2} when ± is minus.

$$x=-\sqrt{15}$$

The equation is now solved.

$$x=\sqrt{15} x=-\sqrt{15}$$