Solve 1/x+7+6/x^2-x-56=x-6/x+7

$$\frac { 1 } { x + 7 } + \frac { 6 } { x ^ { 2 } - x - 56 } = \frac { x - 6 } { x + 7 }$$

Solve for x

$$x=5$$

$$x=10$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -7,8 since division by zero is not defined. Multiply both sides of the equation by \left(x-8\right)\left(x+7\right), the least common multiple of x+7,x^{2}-x-56.

$$x-8+6=\left(x-8\right)\left(x-6\right)$$

Add -8 and 6 to get -2.

$$x-2=\left(x-8\right)\left(x-6\right)$$

Use the distributive property to multiply x-8 by x-6 and combine like terms.

$$x-2=x^{2}-14x+48$$

Subtract x^{2} from both sides.

$$x-2-x^{2}=-14x+48$$

Add 14x to both sides.

$$x-2-x^{2}+14x=48$$

Combine x and 14x to get 15x.

$$15x-2-x^{2}=48$$

Subtract 48 from both sides.

$$15x-2-x^{2}-48=0$$

Subtract 48 from -2 to get -50.

$$15x-50-x^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-x^{2}+15x-50=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-50. To find a and b, set up a system to be solved.

$$a+b=15 ab=-\left(-50\right)=50$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 50.

$$1,50 2,25 5,10$$

Calculate the sum for each pair.

$$1+50=51 2+25=27 5+10=15$$

The solution is the pair that gives sum 15.

$$a=10 b=5$$

Rewrite -x^{2}+15x-50 as \left(-x^{2}+10x\right)+\left(5x-50\right).

$$\left(-x^{2}+10x\right)+\left(5x-50\right)$$

Factor out -x in the first and 5 in the second group.

$$-x\left(x-10\right)+5\left(x-10\right)$$

Factor out common term x-10 by using distributive property.

$$\left(x-10\right)\left(-x+5\right)$$

To find equation solutions, solve x-10=0 and -x+5=0.

$$x=10 x=5$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -7,8 since division by zero is not defined. Multiply both sides of the equation by \left(x-8\right)\left(x+7\right), the least common multiple of x+7,x^{2}-x-56.

$$x-8+6=\left(x-8\right)\left(x-6\right)$$

Add -8 and 6 to get -2.

$$x-2=\left(x-8\right)\left(x-6\right)$$

Use the distributive property to multiply x-8 by x-6 and combine like terms.

$$x-2=x^{2}-14x+48$$

Subtract x^{2} from both sides.

$$x-2-x^{2}=-14x+48$$

Add 14x to both sides.

$$x-2-x^{2}+14x=48$$

Combine x and 14x to get 15x.

$$15x-2-x^{2}=48$$

Subtract 48 from both sides.

$$15x-2-x^{2}-48=0$$

Subtract 48 from -2 to get -50.

$$15x-50-x^{2}=0$$

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.

$$-x^{2}+15x-50=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 15 for b, and -50 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-15±\sqrt{15^{2}-4\left(-1\right)\left(-50\right)}}{2\left(-1\right)}$$

Square 15.

$$x=\frac{-15±\sqrt{225-4\left(-1\right)\left(-50\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-15±\sqrt{225+4\left(-50\right)}}{2\left(-1\right)}$$

Multiply 4 times -50.

$$x=\frac{-15±\sqrt{225-200}}{2\left(-1\right)}$$

Add 225 to -200.

$$x=\frac{-15±\sqrt{25}}{2\left(-1\right)}$$

Take the square root of 25.

$$x=\frac{-15±5}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-15±5}{-2}$$

Now solve the equation x=\frac{-15±5}{-2} when ± is plus. Add -15 to 5.

$$x=-\frac{10}{-2}$$

Divide -10 by -2.

$$x=5$$

Now solve the equation x=\frac{-15±5}{-2} when ± is minus. Subtract 5 from -15.

$$x=-\frac{20}{-2}$$

Divide -20 by -2.

$$x=10$$

The equation is now solved.

$$x=5 x=10$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -7,8 since division by zero is not defined. Multiply both sides of the equation by \left(x-8\right)\left(x+7\right), the least common multiple of x+7,x^{2}-x-56.

$$x-8+6=\left(x-8\right)\left(x-6\right)$$

Add -8 and 6 to get -2.

$$x-2=\left(x-8\right)\left(x-6\right)$$

Use the distributive property to multiply x-8 by x-6 and combine like terms.

$$x-2=x^{2}-14x+48$$

Subtract x^{2} from both sides.

$$x-2-x^{2}=-14x+48$$

Add 14x to both sides.

$$x-2-x^{2}+14x=48$$

Combine x and 14x to get 15x.

$$15x-2-x^{2}=48$$

Add 2 to both sides.

$$15x-x^{2}=48+2$$

Add 48 and 2 to get 50.

$$15x-x^{2}=50$$

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.

$$-x^{2}+15x=50$$

Divide both sides by -1.

$$\frac{-x^{2}+15x}{-1}=\frac{50}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{15}{-1}x=\frac{50}{-1}$$

Divide 15 by -1.

$$x^{2}-15x=\frac{50}{-1}$$

Divide 50 by -1.

$$x^{2}-15x=-50$$

Divide -15, the coefficient of the x term, by 2 to get -\frac{15}{2}. Then add the square of -\frac{15}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-15x+\left(-\frac{15}{2}\right)^{2}=-50+\left(-\frac{15}{2}\right)^{2}$$

Square -\frac{15}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-15x+\frac{225}{4}=-50+\frac{225}{4}$$

Add -50 to \frac{225}{4}.

$$x^{2}-15x+\frac{225}{4}=\frac{25}{4}$$

Factor x^{2}-15x+\frac{225}{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}.

$$\left(x-\frac{15}{2}\right)^{2}=\frac{25}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{15}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Simplify.

$$x-\frac{15}{2}=\frac{5}{2} x-\frac{15}{2}=-\frac{5}{2}$$

Add \frac{15}{2} to both sides of the equation.

$$x=10 x=5$$