Solve 40/x+3-6/x-3=2

$$\frac { 40 } { x + 3 } - \frac { 6 } { x - 3 } = 2$$

Solve for x

$$x=5$$

$$x=12$$

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.

$$\left(x-3\right)\times 40-\left(x+3\right)\times 6=2\left(x-3\right)\left(x+3\right)$$

Use the distributive property to multiply x-3 by 40.

$$40x-120-\left(x+3\right)\times 6=2\left(x-3\right)\left(x+3\right)$$

Use the distributive property to multiply x+3 by 6.

$$40x-120-\left(6x+18\right)=2\left(x-3\right)\left(x+3\right)$$

To find the opposite of 6x+18, find the opposite of each term.

$$40x-120-6x-18=2\left(x-3\right)\left(x+3\right)$$

Combine 40x and -6x to get 34x.

$$34x-120-18=2\left(x-3\right)\left(x+3\right)$$

Subtract 18 from -120 to get -138.

$$34x-138=2\left(x-3\right)\left(x+3\right)$$

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

$$34x-138=\left(2x-6\right)\left(x+3\right)$$

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

$$34x-138=2x^{2}-18$$

Subtract 2x^{2} from both sides.

$$34x-138-2x^{2}=-18$$

Add 18 to both sides.

$$34x-138-2x^{2}+18=0$$

Add -138 and 18 to get -120.

$$34x-120-2x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-2x^{2}+34x-120=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 34 for b, and -120 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-34±\sqrt{34^{2}-4\left(-2\right)\left(-120\right)}}{2\left(-2\right)}$$

Square 34.

$$x=\frac{-34±\sqrt{1156-4\left(-2\right)\left(-120\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-34±\sqrt{1156+8\left(-120\right)}}{2\left(-2\right)}$$

Multiply 8 times -120.

$$x=\frac{-34±\sqrt{1156-960}}{2\left(-2\right)}$$

Add 1156 to -960.

$$x=\frac{-34±\sqrt{196}}{2\left(-2\right)}$$

Take the square root of 196.

$$x=\frac{-34±14}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-34±14}{-4}$$

Now solve the equation x=\frac{-34±14}{-4} when ± is plus. Add -34 to 14.

$$x=-\frac{20}{-4}$$

Divide -20 by -4.

$$x=5$$

Now solve the equation x=\frac{-34±14}{-4} when ± is minus. Subtract 14 from -34.

$$x=-\frac{48}{-4}$$

Divide -48 by -4.

$$x=12$$

The equation is now solved.

$$x=5 x=12$$

Steps for Completing the Square

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.

$$\left(x-3\right)\times 40-\left(x+3\right)\times 6=2\left(x-3\right)\left(x+3\right)$$

Use the distributive property to multiply x-3 by 40.

$$40x-120-\left(x+3\right)\times 6=2\left(x-3\right)\left(x+3\right)$$

Use the distributive property to multiply x+3 by 6.

$$40x-120-\left(6x+18\right)=2\left(x-3\right)\left(x+3\right)$$

To find the opposite of 6x+18, find the opposite of each term.

$$40x-120-6x-18=2\left(x-3\right)\left(x+3\right)$$

Combine 40x and -6x to get 34x.

$$34x-120-18=2\left(x-3\right)\left(x+3\right)$$

Subtract 18 from -120 to get -138.

$$34x-138=2\left(x-3\right)\left(x+3\right)$$

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

$$34x-138=\left(2x-6\right)\left(x+3\right)$$

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

$$34x-138=2x^{2}-18$$

Subtract 2x^{2} from both sides.

$$34x-138-2x^{2}=-18$$

Add 138 to both sides.

$$34x-2x^{2}=-18+138$$

Add -18 and 138 to get 120.

$$34x-2x^{2}=120$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-2x^{2}+34x=120$$

Divide both sides by -2.

$$\frac{-2x^{2}+34x}{-2}=\frac{120}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{34}{-2}x=\frac{120}{-2}$$

Divide 34 by -2.

$$x^{2}-17x=\frac{120}{-2}$$

Divide 120 by -2.

$$x^{2}-17x=-60$$

Divide -17, the coefficient of the x term, by 2 to get -\frac{17}{2}. Then add the square of -\frac{17}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-17x+\left(-\frac{17}{2}\right)^{2}=-60+\left(-\frac{17}{2}\right)^{2}$$

Square -\frac{17}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-17x+\frac{289}{4}=-60+\frac{289}{4}$$

Add -60 to \frac{289}{4}.

$$x^{2}-17x+\frac{289}{4}=\frac{49}{4}$$

Factor x^{2}-17x+\frac{289}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{17}{2}\right)^{2}=\frac{49}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{17}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Simplify.

$$x-\frac{17}{2}=\frac{7}{2} x-\frac{17}{2}=-\frac{7}{2}$$

Add \frac{17}{2} to both sides of the equation.

$$x=12 x=5$$