Solve 3/3-x+2/5+x-1/15=6/3-x

$$\frac{ 3 }{ 3-x } + \frac{ 2 }{ 5+x } - \frac{ 1 }{ 15 } = \frac{ 6 }{ 3-x }$$

Solve for x

$$x=-2$$

$$x=75$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -5,3 since division by zero is not defined. Multiply both sides of the equation by 15\left(x-3\right)\left(x+5\right), the least common multiple of 3-x,5+x,15.

$$\left(-75-15x\right)\times 3+\left(15x-45\right)\times 2+15\left(x-3\right)\left(x+5\right)\left(-\frac{1}{15}\right)=\left(-75-15x\right)\times 6$$

Use the distributive property to multiply -75-15x by 3.

$$-225-45x+\left(15x-45\right)\times 2+15\left(x-3\right)\left(x+5\right)\left(-\frac{1}{15}\right)=\left(-75-15x\right)\times 6$$

Use the distributive property to multiply 15x-45 by 2.

$$-225-45x+30x-90+15\left(x-3\right)\left(x+5\right)\left(-\frac{1}{15}\right)=\left(-75-15x\right)\times 6$$

Combine -45x and 30x to get -15x.

$$-225-15x-90+15\left(x-3\right)\left(x+5\right)\left(-\frac{1}{15}\right)=\left(-75-15x\right)\times 6$$

Subtract 90 from -225 to get -315.

$$-315-15x+15\left(x-3\right)\left(x+5\right)\left(-\frac{1}{15}\right)=\left(-75-15x\right)\times 6$$

Multiply 15 and -\frac{1}{15} to get -1.

$$-315-15x-\left(x-3\right)\left(x+5\right)=\left(-75-15x\right)\times 6$$

Use the distributive property to multiply -1 by x-3.

$$-315-15x+\left(-x+3\right)\left(x+5\right)=\left(-75-15x\right)\times 6$$

Use the distributive property to multiply -x+3 by x+5 and combine like terms.

$$-315-15x-x^{2}-2x+15=\left(-75-15x\right)\times 6$$

Combine -15x and -2x to get -17x.

$$-315-17x-x^{2}+15=\left(-75-15x\right)\times 6$$

Add -315 and 15 to get -300.

$$-300-17x-x^{2}=\left(-75-15x\right)\times 6$$

Use the distributive property to multiply -75-15x by 6.

$$-300-17x-x^{2}=-450-90x$$

Subtract -450 from both sides.

$$-300-17x-x^{2}-\left(-450\right)=-90x$$

The opposite of -450 is 450.

$$-300-17x-x^{2}+450=-90x$$

Add 90x to both sides.

$$-300-17x-x^{2}+450+90x=0$$

Add -300 and 450 to get 150.

$$150-17x-x^{2}+90x=0$$

Combine -17x and 90x to get 73x.

$$150+73x-x^{2}=0$$

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.

$$-x^{2}+73x+150=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 73 for b, and 150 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-73±\sqrt{73^{2}-4\left(-1\right)\times 150}}{2\left(-1\right)}$$

Square 73.

$$x=\frac{-73±\sqrt{5329-4\left(-1\right)\times 150}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-73±\sqrt{5329+4\times 150}}{2\left(-1\right)}$$

Multiply 4 times 150.

$$x=\frac{-73±\sqrt{5329+600}}{2\left(-1\right)}$$

Add 5329 to 600.

$$x=\frac{-73±\sqrt{5929}}{2\left(-1\right)}$$

Take the square root of 5929.

$$x=\frac{-73±77}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-73±77}{-2}$$

Now solve the equation x=\frac{-73±77}{-2} when ± is plus. Add -73 to 77.

$$x=\frac{4}{-2}$$

Divide 4 by -2.

$$x=-2$$

Now solve the equation x=\frac{-73±77}{-2} when ± is minus. Subtract 77 from -73.

$$x=-\frac{150}{-2}$$

Divide -150 by -2.

$$x=75$$

The equation is now solved.

$$x=-2 x=75$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -5,3 since division by zero is not defined. Multiply both sides of the equation by 15\left(x-3\right)\left(x+5\right), the least common multiple of 3-x,5+x,15.

$$\left(-75-15x\right)\times 3+\left(15x-45\right)\times 2+15\left(x-3\right)\left(x+5\right)\left(-\frac{1}{15}\right)=\left(-75-15x\right)\times 6$$

Use the distributive property to multiply -75-15x by 3.

$$-225-45x+\left(15x-45\right)\times 2+15\left(x-3\right)\left(x+5\right)\left(-\frac{1}{15}\right)=\left(-75-15x\right)\times 6$$

Use the distributive property to multiply 15x-45 by 2.

$$-225-45x+30x-90+15\left(x-3\right)\left(x+5\right)\left(-\frac{1}{15}\right)=\left(-75-15x\right)\times 6$$

Combine -45x and 30x to get -15x.

$$-225-15x-90+15\left(x-3\right)\left(x+5\right)\left(-\frac{1}{15}\right)=\left(-75-15x\right)\times 6$$

Subtract 90 from -225 to get -315.

$$-315-15x+15\left(x-3\right)\left(x+5\right)\left(-\frac{1}{15}\right)=\left(-75-15x\right)\times 6$$

Multiply 15 and -\frac{1}{15} to get -1.

$$-315-15x-\left(x-3\right)\left(x+5\right)=\left(-75-15x\right)\times 6$$

Use the distributive property to multiply -1 by x-3.

$$-315-15x+\left(-x+3\right)\left(x+5\right)=\left(-75-15x\right)\times 6$$

Use the distributive property to multiply -x+3 by x+5 and combine like terms.

$$-315-15x-x^{2}-2x+15=\left(-75-15x\right)\times 6$$

Combine -15x and -2x to get -17x.

$$-315-17x-x^{2}+15=\left(-75-15x\right)\times 6$$

Add -315 and 15 to get -300.

$$-300-17x-x^{2}=\left(-75-15x\right)\times 6$$

Use the distributive property to multiply -75-15x by 6.

$$-300-17x-x^{2}=-450-90x$$

Add 90x to both sides.

$$-300-17x-x^{2}+90x=-450$$

Combine -17x and 90x to get 73x.

$$-300+73x-x^{2}=-450$$

Add 300 to both sides.

$$73x-x^{2}=-450+300$$

Add -450 and 300 to get -150.

$$73x-x^{2}=-150$$

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.

$$-x^{2}+73x=-150$$

Divide both sides by -1.

$$\frac{-x^{2}+73x}{-1}=-\frac{150}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{73}{-1}x=-\frac{150}{-1}$$

Divide 73 by -1.

$$x^{2}-73x=-\frac{150}{-1}$$

Divide -150 by -1.

$$x^{2}-73x=150$$

Divide -73, the coefficient of the x term, by 2 to get -\frac{73}{2}. Then add the square of -\frac{73}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-73x+\left(-\frac{73}{2}\right)^{2}=150+\left(-\frac{73}{2}\right)^{2}$$

Square -\frac{73}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-73x+\frac{5329}{4}=150+\frac{5329}{4}$$

Add 150 to \frac{5329}{4}.

$$x^{2}-73x+\frac{5329}{4}=\frac{5929}{4}$$

Factor x^{2}-73x+\frac{5329}{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}.

$$\left(x-\frac{73}{2}\right)^{2}=\frac{5929}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{73}{2}\right)^{2}}=\sqrt{\frac{5929}{4}}$$

Simplify.

$$x-\frac{73}{2}=\frac{77}{2} x-\frac{73}{2}=-\frac{77}{2}$$

Add \frac{73}{2} to both sides of the equation.

$$x=75 x=-2$$