Solve 8/x+5-5-3x/x^2-25=7/5x

$$\frac { 8 } { x + 5 } - \frac { 5 - 3 x } { x ^ { 2 } - 25 } = \frac { 7 } { 5 x }$$

Solve for x

$$x=\frac{5\sqrt{681}}{96}+\frac{75}{32}\approx 3.702915453$$

$$x=-\frac{5\sqrt{681}}{96}+\frac{75}{32}\approx 0.984584547$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -5,0,5 since division by zero is not defined. Multiply both sides of the equation by 5x\left(x-5\right)\left(x+5\right), the least common multiple of x+5,x^{2}-25,5x.

$$5x\left(x-5\right)\times 8-5x\left(5-3x\right)=\left(x^{2}-25\right)\times 7$$

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

$$\left(5x^{2}-25x\right)\times 8-5x\left(5-3x\right)=\left(x^{2}-25\right)\times 7$$

Use the distributive property to multiply 5x^{2}-25x by 8.

$$40x^{2}-200x-5x\left(5-3x\right)=\left(x^{2}-25\right)\times 7$$

Use the distributive property to multiply 5x by 5-3x.

$$40x^{2}-200x-\left(25x-15x^{2}\right)=\left(x^{2}-25\right)\times 7$$

To find the opposite of 25x-15x^{2}, find the opposite of each term.

$$40x^{2}-200x-25x+15x^{2}=\left(x^{2}-25\right)\times 7$$

Combine -200x and -25x to get -225x.

$$40x^{2}-225x+15x^{2}=\left(x^{2}-25\right)\times 7$$

Combine 40x^{2} and 15x^{2} to get 55x^{2}.

$$55x^{2}-225x=\left(x^{2}-25\right)\times 7$$

Use the distributive property to multiply x^{2}-25 by 7.

$$55x^{2}-225x=7x^{2}-175$$

Subtract 7x^{2} from both sides.

$$55x^{2}-225x-7x^{2}=-175$$

Combine 55x^{2} and -7x^{2} to get 48x^{2}.

$$48x^{2}-225x=-175$$

Add 175 to both sides.

$$48x^{2}-225x+175=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 48 for a, -225 for b, and 175 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-225\right)±\sqrt{\left(-225\right)^{2}-4\times 48\times 175}}{2\times 48}$$

Square -225.

$$x=\frac{-\left(-225\right)±\sqrt{50625-4\times 48\times 175}}{2\times 48}$$

Multiply -4 times 48.

$$x=\frac{-\left(-225\right)±\sqrt{50625-192\times 175}}{2\times 48}$$

Multiply -192 times 175.

$$x=\frac{-\left(-225\right)±\sqrt{50625-33600}}{2\times 48}$$

Add 50625 to -33600.

$$x=\frac{-\left(-225\right)±\sqrt{17025}}{2\times 48}$$

Take the square root of 17025.

$$x=\frac{-\left(-225\right)±5\sqrt{681}}{2\times 48}$$

The opposite of -225 is 225.

$$x=\frac{225±5\sqrt{681}}{2\times 48}$$

Multiply 2 times 48.

$$x=\frac{225±5\sqrt{681}}{96}$$

Now solve the equation x=\frac{225±5\sqrt{681}}{96} when ± is plus. Add 225 to 5\sqrt{681}.

$$x=\frac{5\sqrt{681}+225}{96}$$

Divide 225+5\sqrt{681} by 96.

$$x=\frac{5\sqrt{681}}{96}+\frac{75}{32}$$

Now solve the equation x=\frac{225±5\sqrt{681}}{96} when ± is minus. Subtract 5\sqrt{681} from 225.

$$x=\frac{225-5\sqrt{681}}{96}$$

Divide 225-5\sqrt{681} by 96.

$$x=-\frac{5\sqrt{681}}{96}+\frac{75}{32}$$

The equation is now solved.

$$x=\frac{5\sqrt{681}}{96}+\frac{75}{32} x=-\frac{5\sqrt{681}}{96}+\frac{75}{32}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -5,0,5 since division by zero is not defined. Multiply both sides of the equation by 5x\left(x-5\right)\left(x+5\right), the least common multiple of x+5,x^{2}-25,5x.

$$5x\left(x-5\right)\times 8-5x\left(5-3x\right)=\left(x^{2}-25\right)\times 7$$

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

$$\left(5x^{2}-25x\right)\times 8-5x\left(5-3x\right)=\left(x^{2}-25\right)\times 7$$

Use the distributive property to multiply 5x^{2}-25x by 8.

$$40x^{2}-200x-5x\left(5-3x\right)=\left(x^{2}-25\right)\times 7$$

Use the distributive property to multiply 5x by 5-3x.

$$40x^{2}-200x-\left(25x-15x^{2}\right)=\left(x^{2}-25\right)\times 7$$

To find the opposite of 25x-15x^{2}, find the opposite of each term.

$$40x^{2}-200x-25x+15x^{2}=\left(x^{2}-25\right)\times 7$$

Combine -200x and -25x to get -225x.

$$40x^{2}-225x+15x^{2}=\left(x^{2}-25\right)\times 7$$

Combine 40x^{2} and 15x^{2} to get 55x^{2}.

$$55x^{2}-225x=\left(x^{2}-25\right)\times 7$$

Use the distributive property to multiply x^{2}-25 by 7.

$$55x^{2}-225x=7x^{2}-175$$

Subtract 7x^{2} from both sides.

$$55x^{2}-225x-7x^{2}=-175$$

Combine 55x^{2} and -7x^{2} to get 48x^{2}.

$$48x^{2}-225x=-175$$

Divide both sides by 48.

$$\frac{48x^{2}-225x}{48}=-\frac{175}{48}$$

Dividing by 48 undoes the multiplication by 48.

$$x^{2}+\left(-\frac{225}{48}\right)x=-\frac{175}{48}$$

Reduce the fraction \frac{-225}{48} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{75}{16}x=-\frac{175}{48}$$

Divide -\frac{75}{16}, the coefficient of the x term, by 2 to get -\frac{75}{32}. Then add the square of -\frac{75}{32} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{75}{16}x+\left(-\frac{75}{32}\right)^{2}=-\frac{175}{48}+\left(-\frac{75}{32}\right)^{2}$$

Square -\frac{75}{32} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{75}{16}x+\frac{5625}{1024}=-\frac{175}{48}+\frac{5625}{1024}$$

Add -\frac{175}{48} to \frac{5625}{1024} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{75}{16}x+\frac{5625}{1024}=\frac{5675}{3072}$$

Factor x^{2}-\frac{75}{16}x+\frac{5625}{1024}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{75}{32}\right)^{2}=\frac{5675}{3072}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{75}{32}\right)^{2}}=\sqrt{\frac{5675}{3072}}$$

Simplify.

$$x-\frac{75}{32}=\frac{5\sqrt{681}}{96} x-\frac{75}{32}=-\frac{5\sqrt{681}}{96}$$

Add \frac{75}{32} to both sides of the equation.

$$x=\frac{5\sqrt{681}}{96}+\frac{75}{32} x=-\frac{5\sqrt{681}}{96}+\frac{75}{32}$$