Solve 5/x+4-2/x-10=3/5

$$\frac{ 5 }{ x+4 } - \frac{ 2 }{ x-10 } = \frac{ 3 }{ 5 }$$

Solve for x (complex solution)

$$x=-\frac{\sqrt{951}i}{6}+\frac{11}{2}\approx 5.5-5.139714648i$$

$$x=\frac{\sqrt{951}i}{6}+\frac{11}{2}\approx 5.5+5.139714648i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -4,10 since division by zero is not defined. Multiply both sides of the equation by 5\left(x-10\right)\left(x+4\right), the least common multiple of x+4,x-10,5.

$$\left(5x-50\right)\times 5-\left(5x+20\right)\times 2=3\left(x-10\right)\left(x+4\right)$$

Use the distributive property to multiply 5x-50 by 5.

$$25x-250-\left(5x+20\right)\times 2=3\left(x-10\right)\left(x+4\right)$$

Use the distributive property to multiply 5x+20 by 2.

$$25x-250-\left(10x+40\right)=3\left(x-10\right)\left(x+4\right)$$

To find the opposite of 10x+40, find the opposite of each term.

$$25x-250-10x-40=3\left(x-10\right)\left(x+4\right)$$

Combine 25x and -10x to get 15x.

$$15x-250-40=3\left(x-10\right)\left(x+4\right)$$

Subtract 40 from -250 to get -290.

$$15x-290=3\left(x-10\right)\left(x+4\right)$$

Use the distributive property to multiply 3 by x-10.

$$15x-290=\left(3x-30\right)\left(x+4\right)$$

Use the distributive property to multiply 3x-30 by x+4 and combine like terms.

$$15x-290=3x^{2}-18x-120$$

Subtract 3x^{2} from both sides.

$$15x-290-3x^{2}=-18x-120$$

Add 18x to both sides.

$$15x-290-3x^{2}+18x=-120$$

Combine 15x and 18x to get 33x.

$$33x-290-3x^{2}=-120$$

Add 120 to both sides.

$$33x-290-3x^{2}+120=0$$

Add -290 and 120 to get -170.

$$33x-170-3x^{2}=0$$

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.

$$-3x^{2}+33x-170=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 33 for b, and -170 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-33±\sqrt{33^{2}-4\left(-3\right)\left(-170\right)}}{2\left(-3\right)}$$

Square 33.

$$x=\frac{-33±\sqrt{1089-4\left(-3\right)\left(-170\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-33±\sqrt{1089+12\left(-170\right)}}{2\left(-3\right)}$$

Multiply 12 times -170.

$$x=\frac{-33±\sqrt{1089-2040}}{2\left(-3\right)}$$

Add 1089 to -2040.

$$x=\frac{-33±\sqrt{-951}}{2\left(-3\right)}$$

Take the square root of -951.

$$x=\frac{-33±\sqrt{951}i}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-33±\sqrt{951}i}{-6}$$

Now solve the equation x=\frac{-33±\sqrt{951}i}{-6} when ± is plus. Add -33 to i\sqrt{951}.

$$x=\frac{-33+\sqrt{951}i}{-6}$$

Divide -33+i\sqrt{951} by -6.

$$x=-\frac{\sqrt{951}i}{6}+\frac{11}{2}$$

Now solve the equation x=\frac{-33±\sqrt{951}i}{-6} when ± is minus. Subtract i\sqrt{951} from -33.

$$x=\frac{-\sqrt{951}i-33}{-6}$$

Divide -33-i\sqrt{951} by -6.

$$x=\frac{\sqrt{951}i}{6}+\frac{11}{2}$$

The equation is now solved.

$$x=-\frac{\sqrt{951}i}{6}+\frac{11}{2} x=\frac{\sqrt{951}i}{6}+\frac{11}{2}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -4,10 since division by zero is not defined. Multiply both sides of the equation by 5\left(x-10\right)\left(x+4\right), the least common multiple of x+4,x-10,5.

$$\left(5x-50\right)\times 5-\left(5x+20\right)\times 2=3\left(x-10\right)\left(x+4\right)$$

Use the distributive property to multiply 5x-50 by 5.

$$25x-250-\left(5x+20\right)\times 2=3\left(x-10\right)\left(x+4\right)$$

Use the distributive property to multiply 5x+20 by 2.

$$25x-250-\left(10x+40\right)=3\left(x-10\right)\left(x+4\right)$$

To find the opposite of 10x+40, find the opposite of each term.

$$25x-250-10x-40=3\left(x-10\right)\left(x+4\right)$$

Combine 25x and -10x to get 15x.

$$15x-250-40=3\left(x-10\right)\left(x+4\right)$$

Subtract 40 from -250 to get -290.

$$15x-290=3\left(x-10\right)\left(x+4\right)$$

Use the distributive property to multiply 3 by x-10.

$$15x-290=\left(3x-30\right)\left(x+4\right)$$

Use the distributive property to multiply 3x-30 by x+4 and combine like terms.

$$15x-290=3x^{2}-18x-120$$

Subtract 3x^{2} from both sides.

$$15x-290-3x^{2}=-18x-120$$

Add 18x to both sides.

$$15x-290-3x^{2}+18x=-120$$

Combine 15x and 18x to get 33x.

$$33x-290-3x^{2}=-120$$

Add 290 to both sides.

$$33x-3x^{2}=-120+290$$

Add -120 and 290 to get 170.

$$33x-3x^{2}=170$$

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.

$$-3x^{2}+33x=170$$

Divide both sides by -3.

$$\frac{-3x^{2}+33x}{-3}=\frac{170}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\frac{33}{-3}x=\frac{170}{-3}$$

Divide 33 by -3.

$$x^{2}-11x=\frac{170}{-3}$$

Divide 170 by -3.

$$x^{2}-11x=-\frac{170}{3}$$

Divide -11, the coefficient of the x term, by 2 to get -\frac{11}{2}. Then add the square of -\frac{11}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-11x+\left(-\frac{11}{2}\right)^{2}=-\frac{170}{3}+\left(-\frac{11}{2}\right)^{2}$$

Square -\frac{11}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-11x+\frac{121}{4}=-\frac{170}{3}+\frac{121}{4}$$

Add -\frac{170}{3} to \frac{121}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-11x+\frac{121}{4}=-\frac{317}{12}$$

Factor x^{2}-11x+\frac{121}{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}.

$$\left(x-\frac{11}{2}\right)^{2}=-\frac{317}{12}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{11}{2}\right)^{2}}=\sqrt{-\frac{317}{12}}$$

Simplify.

$$x-\frac{11}{2}=\frac{\sqrt{951}i}{6} x-\frac{11}{2}=-\frac{\sqrt{951}i}{6}$$

Add \frac{11}{2} to both sides of the equation.

$$x=\frac{\sqrt{951}i}{6}+\frac{11}{2} x=-\frac{\sqrt{951}i}{6}+\frac{11}{2}$$