Solve 54/15-x-5/15+x=3

$$\frac{ 54 }{ 15-x } - \frac{ 5 }{ 15+x } =3$$

Solve for x

$$x=\frac{\sqrt{2761}-59}{6}\approx -1.075796896$$

$$x=\frac{-\sqrt{2761}-59}{6}\approx -18.59086977$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -15,15 since division by zero is not defined. Multiply both sides of the equation by \left(x-15\right)\left(x+15\right), the least common multiple of 15-x,15+x.

$$-\left(15+x\right)\times 54-\left(x-15\right)\times 5=3\left(x-15\right)\left(x+15\right)$$

To find the opposite of 15+x, find the opposite of each term.

$$\left(-15-x\right)\times 54-\left(x-15\right)\times 5=3\left(x-15\right)\left(x+15\right)$$

Use the distributive property to multiply -15-x by 54.

$$-810-54x-\left(x-15\right)\times 5=3\left(x-15\right)\left(x+15\right)$$

Use the distributive property to multiply x-15 by 5.

$$-810-54x-\left(5x-75\right)=3\left(x-15\right)\left(x+15\right)$$

To find the opposite of 5x-75, find the opposite of each term.

$$-810-54x-5x+75=3\left(x-15\right)\left(x+15\right)$$

Combine -54x and -5x to get -59x.

$$-810-59x+75=3\left(x-15\right)\left(x+15\right)$$

Add -810 and 75 to get -735.

$$-735-59x=3\left(x-15\right)\left(x+15\right)$$

Use the distributive property to multiply 3 by x-15.

$$-735-59x=\left(3x-45\right)\left(x+15\right)$$

Use the distributive property to multiply 3x-45 by x+15 and combine like terms.

$$-735-59x=3x^{2}-675$$

Subtract 3x^{2} from both sides.

$$-735-59x-3x^{2}=-675$$

Add 675 to both sides.

$$-735-59x-3x^{2}+675=0$$

Add -735 and 675 to get -60.

$$-60-59x-3x^{2}=0$$

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.

$$-3x^{2}-59x-60=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, -59 for b, and -60 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-59\right)±\sqrt{\left(-59\right)^{2}-4\left(-3\right)\left(-60\right)}}{2\left(-3\right)}$$

Square -59.

$$x=\frac{-\left(-59\right)±\sqrt{3481-4\left(-3\right)\left(-60\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-\left(-59\right)±\sqrt{3481+12\left(-60\right)}}{2\left(-3\right)}$$

Multiply 12 times -60.

$$x=\frac{-\left(-59\right)±\sqrt{3481-720}}{2\left(-3\right)}$$

Add 3481 to -720.

$$x=\frac{-\left(-59\right)±\sqrt{2761}}{2\left(-3\right)}$$

The opposite of -59 is 59.

$$x=\frac{59±\sqrt{2761}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{59±\sqrt{2761}}{-6}$$

Now solve the equation x=\frac{59±\sqrt{2761}}{-6} when ± is plus. Add 59 to \sqrt{2761}.

$$x=\frac{\sqrt{2761}+59}{-6}$$

Divide 59+\sqrt{2761} by -6.

$$x=\frac{-\sqrt{2761}-59}{6}$$

Now solve the equation x=\frac{59±\sqrt{2761}}{-6} when ± is minus. Subtract \sqrt{2761} from 59.

$$x=\frac{59-\sqrt{2761}}{-6}$$

Divide 59-\sqrt{2761} by -6.

$$x=\frac{\sqrt{2761}-59}{6}$$

The equation is now solved.

$$x=\frac{-\sqrt{2761}-59}{6} x=\frac{\sqrt{2761}-59}{6}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -15,15 since division by zero is not defined. Multiply both sides of the equation by \left(x-15\right)\left(x+15\right), the least common multiple of 15-x,15+x.

$$-\left(15+x\right)\times 54-\left(x-15\right)\times 5=3\left(x-15\right)\left(x+15\right)$$

To find the opposite of 15+x, find the opposite of each term.

$$\left(-15-x\right)\times 54-\left(x-15\right)\times 5=3\left(x-15\right)\left(x+15\right)$$

Use the distributive property to multiply -15-x by 54.

$$-810-54x-\left(x-15\right)\times 5=3\left(x-15\right)\left(x+15\right)$$

Use the distributive property to multiply x-15 by 5.

$$-810-54x-\left(5x-75\right)=3\left(x-15\right)\left(x+15\right)$$

To find the opposite of 5x-75, find the opposite of each term.

$$-810-54x-5x+75=3\left(x-15\right)\left(x+15\right)$$

Combine -54x and -5x to get -59x.

$$-810-59x+75=3\left(x-15\right)\left(x+15\right)$$

Add -810 and 75 to get -735.

$$-735-59x=3\left(x-15\right)\left(x+15\right)$$

Use the distributive property to multiply 3 by x-15.

$$-735-59x=\left(3x-45\right)\left(x+15\right)$$

Use the distributive property to multiply 3x-45 by x+15 and combine like terms.

$$-735-59x=3x^{2}-675$$

Subtract 3x^{2} from both sides.

$$-735-59x-3x^{2}=-675$$

Add 735 to both sides.

$$-59x-3x^{2}=-675+735$$

Add -675 and 735 to get 60.

$$-59x-3x^{2}=60$$

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.

$$-3x^{2}-59x=60$$

Divide both sides by -3.

$$\frac{-3x^{2}-59x}{-3}=\frac{60}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\left(-\frac{59}{-3}\right)x=\frac{60}{-3}$$

Divide -59 by -3.

$$x^{2}+\frac{59}{3}x=\frac{60}{-3}$$

Divide 60 by -3.

$$x^{2}+\frac{59}{3}x=-20$$

Divide \frac{59}{3}, the coefficient of the x term, by 2 to get \frac{59}{6}. Then add the square of \frac{59}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{59}{3}x+\left(\frac{59}{6}\right)^{2}=-20+\left(\frac{59}{6}\right)^{2}$$

Square \frac{59}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{59}{3}x+\frac{3481}{36}=-20+\frac{3481}{36}$$

Add -20 to \frac{3481}{36}.

$$x^{2}+\frac{59}{3}x+\frac{3481}{36}=\frac{2761}{36}$$

Factor x^{2}+\frac{59}{3}x+\frac{3481}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{59}{6}\right)^{2}=\frac{2761}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{59}{6}\right)^{2}}=\sqrt{\frac{2761}{36}}$$

Simplify.

$$x+\frac{59}{6}=\frac{\sqrt{2761}}{6} x+\frac{59}{6}=-\frac{\sqrt{2761}}{6}$$

Subtract \frac{59}{6} from both sides of the equation.

$$x=\frac{\sqrt{2761}-59}{6} x=\frac{-\sqrt{2761}-59}{6}$$