Solve 10/(x+3)(x-7)-8/(x+3)(x-5)=x+10/(x-5)(x-7)

$$\frac { 10 } { ( x + 3 ) ( x - 7 ) } - \frac { 8 } { ( x + 3 ) ( x - 5 ) } = \frac { x + 10 } { ( x - 5 ) ( x - 7 ) }$$

Solve for x

$$x=-8$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -3,5,7 since division by zero is not defined. Multiply both sides of the equation by \left(x-7\right)\left(x-5\right)\left(x+3\right), the least common multiple of \left(x+3\right)\left(x-7\right),\left(x+3\right)\left(x-5\right),\left(x-5\right)\left(x-7\right).

$$\left(x-5\right)\times 10-\left(x-7\right)\times 8=\left(x+3\right)\left(x+10\right)$$

Use the distributive property to multiply x-5 by 10.

$$10x-50-\left(x-7\right)\times 8=\left(x+3\right)\left(x+10\right)$$

Use the distributive property to multiply x-7 by 8.

$$10x-50-\left(8x-56\right)=\left(x+3\right)\left(x+10\right)$$

To find the opposite of 8x-56, find the opposite of each term.

$$10x-50-8x+56=\left(x+3\right)\left(x+10\right)$$

Combine 10x and -8x to get 2x.

$$2x-50+56=\left(x+3\right)\left(x+10\right)$$

Add -50 and 56 to get 6.

$$2x+6=\left(x+3\right)\left(x+10\right)$$

Use the distributive property to multiply x+3 by x+10 and combine like terms.

$$2x+6=x^{2}+13x+30$$

Subtract x^{2} from both sides.

$$2x+6-x^{2}=13x+30$$

Subtract 13x from both sides.

$$2x+6-x^{2}-13x=30$$

Combine 2x and -13x to get -11x.

$$-11x+6-x^{2}=30$$

Subtract 30 from both sides.

$$-11x+6-x^{2}-30=0$$

Subtract 30 from 6 to get -24.

$$-11x-24-x^{2}=0$$

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.

$$-x^{2}-11x-24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -11 for b, and -24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\left(-1\right)\left(-24\right)}}{2\left(-1\right)}$$

Square -11.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\left(-1\right)\left(-24\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-11\right)±\sqrt{121+4\left(-24\right)}}{2\left(-1\right)}$$

Multiply 4 times -24.

$$x=\frac{-\left(-11\right)±\sqrt{121-96}}{2\left(-1\right)}$$

Add 121 to -96.

$$x=\frac{-\left(-11\right)±\sqrt{25}}{2\left(-1\right)}$$

Take the square root of 25.

$$x=\frac{-\left(-11\right)±5}{2\left(-1\right)}$$

The opposite of -11 is 11.

$$x=\frac{11±5}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{11±5}{-2}$$

Now solve the equation x=\frac{11±5}{-2} when ± is plus. Add 11 to 5.

$$x=\frac{16}{-2}$$

Divide 16 by -2.

$$x=-8$$

Now solve the equation x=\frac{11±5}{-2} when ± is minus. Subtract 5 from 11.

$$x=\frac{6}{-2}$$

Divide 6 by -2.

$$x=-3$$

The equation is now solved.

$$x=-8 x=-3$$

Variable x cannot be equal to -3.

$$x=-8$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -3,5,7 since division by zero is not defined. Multiply both sides of the equation by \left(x-7\right)\left(x-5\right)\left(x+3\right), the least common multiple of \left(x+3\right)\left(x-7\right),\left(x+3\right)\left(x-5\right),\left(x-5\right)\left(x-7\right).

$$\left(x-5\right)\times 10-\left(x-7\right)\times 8=\left(x+3\right)\left(x+10\right)$$

Use the distributive property to multiply x-5 by 10.

$$10x-50-\left(x-7\right)\times 8=\left(x+3\right)\left(x+10\right)$$

Use the distributive property to multiply x-7 by 8.

$$10x-50-\left(8x-56\right)=\left(x+3\right)\left(x+10\right)$$

To find the opposite of 8x-56, find the opposite of each term.

$$10x-50-8x+56=\left(x+3\right)\left(x+10\right)$$

Combine 10x and -8x to get 2x.

$$2x-50+56=\left(x+3\right)\left(x+10\right)$$

Add -50 and 56 to get 6.

$$2x+6=\left(x+3\right)\left(x+10\right)$$

Use the distributive property to multiply x+3 by x+10 and combine like terms.

$$2x+6=x^{2}+13x+30$$

Subtract x^{2} from both sides.

$$2x+6-x^{2}=13x+30$$

Subtract 13x from both sides.

$$2x+6-x^{2}-13x=30$$

Combine 2x and -13x to get -11x.

$$-11x+6-x^{2}=30$$

Subtract 6 from both sides.

$$-11x-x^{2}=30-6$$

Subtract 6 from 30 to get 24.

$$-11x-x^{2}=24$$

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.

$$-x^{2}-11x=24$$

Divide both sides by -1.

$$\frac{-x^{2}-11x}{-1}=\frac{24}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{11}{-1}\right)x=\frac{24}{-1}$$

Divide -11 by -1.

$$x^{2}+11x=\frac{24}{-1}$$

Divide 24 by -1.

$$x^{2}+11x=-24$$

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.

$$x^{2}+11x+\left(\frac{11}{2}\right)^{2}=-24+\left(\frac{11}{2}\right)^{2}$$

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

$$x^{2}+11x+\frac{121}{4}=-24+\frac{121}{4}$$

Add -24 to \frac{121}{4}.

$$x^{2}+11x+\frac{121}{4}=\frac{25}{4}$$

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}.

$$\left(x+\frac{11}{2}\right)^{2}=\frac{25}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{11}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Simplify.

$$x+\frac{11}{2}=\frac{5}{2} x+\frac{11}{2}=-\frac{5}{2}$$

Subtract \frac{11}{2} from both sides of the equation.

$$x=-3 x=-8$$

Variable x cannot be equal to -3.

$$x=-8$$