Solve 7x+7/x+2+3x+3/4-x=2-x^2+26x+6/x^2-2x-8

$$\frac { 7 x + 7 } { x + 2 } + \frac { 3 x + 3 } { 4 - x } = 2 - \frac { x ^ { 2 } + 26 x + 6 } { x ^ { 2 } - 2 x - 8 }$$

Solve for x

$$x=2$$

Steps Using Difference of Squares

Variable x cannot be equal to any of the values -2,4 since division by zero is not defined. Multiply both sides of the equation by \left(x-4\right)\left(x+2\right), the least common multiple of x+2,4-x,x^{2}-2x-8.

$$\left(x-4\right)\left(7x+7\right)-\left(2+x\right)\left(3x+3\right)=\left(x-4\right)\left(x+2\right)\times 2-\left(x^{2}+26x+6\right)$$

Use the distributive property to multiply x-4 by 7x+7 and combine like terms.

$$7x^{2}-21x-28-\left(2+x\right)\left(3x+3\right)=\left(x-4\right)\left(x+2\right)\times 2-\left(x^{2}+26x+6\right)$$

Use the distributive property to multiply -1 by 2+x.

$$7x^{2}-21x-28+\left(-2-x\right)\left(3x+3\right)=\left(x-4\right)\left(x+2\right)\times 2-\left(x^{2}+26x+6\right)$$

Use the distributive property to multiply -2-x by 3x+3 and combine like terms.

$$7x^{2}-21x-28-9x-6-3x^{2}=\left(x-4\right)\left(x+2\right)\times 2-\left(x^{2}+26x+6\right)$$

Combine -21x and -9x to get -30x.

$$7x^{2}-30x-28-6-3x^{2}=\left(x-4\right)\left(x+2\right)\times 2-\left(x^{2}+26x+6\right)$$

Subtract 6 from -28 to get -34.

$$7x^{2}-30x-34-3x^{2}=\left(x-4\right)\left(x+2\right)\times 2-\left(x^{2}+26x+6\right)$$

Combine 7x^{2} and -3x^{2} to get 4x^{2}.

$$4x^{2}-30x-34=\left(x-4\right)\left(x+2\right)\times 2-\left(x^{2}+26x+6\right)$$

Use the distributive property to multiply x-4 by x+2 and combine like terms.

$$4x^{2}-30x-34=\left(x^{2}-2x-8\right)\times 2-\left(x^{2}+26x+6\right)$$

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

$$4x^{2}-30x-34=2x^{2}-4x-16-\left(x^{2}+26x+6\right)$$

To find the opposite of x^{2}+26x+6, find the opposite of each term.

$$4x^{2}-30x-34=2x^{2}-4x-16-x^{2}-26x-6$$

Combine 2x^{2} and -x^{2} to get x^{2}.

$$4x^{2}-30x-34=x^{2}-4x-16-26x-6$$

Combine -4x and -26x to get -30x.

$$4x^{2}-30x-34=x^{2}-30x-16-6$$

Subtract 6 from -16 to get -22.

$$4x^{2}-30x-34=x^{2}-30x-22$$

Subtract x^{2} from both sides.

$$4x^{2}-30x-34-x^{2}=-30x-22$$

Combine 4x^{2} and -x^{2} to get 3x^{2}.

$$3x^{2}-30x-34=-30x-22$$

Add 30x to both sides.

$$3x^{2}-30x-34+30x=-22$$

Combine -30x and 30x to get 0.

$$3x^{2}-34=-22$$

Add 22 to both sides.

$$3x^{2}-34+22=0$$

Add -34 and 22 to get -12.

$$3x^{2}-12=0$$

Divide both sides by 3.

$$x^{2}-4=0$$

Consider x^{2}-4. Rewrite x^{2}-4 as x^{2}-2^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x-2\right)\left(x+2\right)=0$$

To find equation solutions, solve x-2=0 and x+2=0.

$$x=2 x=-2$$

Variable x cannot be equal to -2.

$$x=2$$

Steps by Finding Square Root

Variable x cannot be equal to any of the values -2,4 since division by zero is not defined. Multiply both sides of the equation by \left(x-4\right)\left(x+2\right), the least common multiple of x+2,4-x,x^{2}-2x-8.

$$\left(x-4\right)\left(7x+7\right)-\left(2+x\right)\left(3x+3\right)=\left(x-4\right)\left(x+2\right)\times 2-\left(x^{2}+26x+6\right)$$

Use the distributive property to multiply x-4 by 7x+7 and combine like terms.

$$7x^{2}-21x-28-\left(2+x\right)\left(3x+3\right)=\left(x-4\right)\left(x+2\right)\times 2-\left(x^{2}+26x+6\right)$$

Use the distributive property to multiply -1 by 2+x.

$$7x^{2}-21x-28+\left(-2-x\right)\left(3x+3\right)=\left(x-4\right)\left(x+2\right)\times 2-\left(x^{2}+26x+6\right)$$

Use the distributive property to multiply -2-x by 3x+3 and combine like terms.

$$7x^{2}-21x-28-9x-6-3x^{2}=\left(x-4\right)\left(x+2\right)\times 2-\left(x^{2}+26x+6\right)$$

Combine -21x and -9x to get -30x.

$$7x^{2}-30x-28-6-3x^{2}=\left(x-4\right)\left(x+2\right)\times 2-\left(x^{2}+26x+6\right)$$

Subtract 6 from -28 to get -34.

$$7x^{2}-30x-34-3x^{2}=\left(x-4\right)\left(x+2\right)\times 2-\left(x^{2}+26x+6\right)$$

Combine 7x^{2} and -3x^{2} to get 4x^{2}.

$$4x^{2}-30x-34=\left(x-4\right)\left(x+2\right)\times 2-\left(x^{2}+26x+6\right)$$

Use the distributive property to multiply x-4 by x+2 and combine like terms.

$$4x^{2}-30x-34=\left(x^{2}-2x-8\right)\times 2-\left(x^{2}+26x+6\right)$$

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

$$4x^{2}-30x-34=2x^{2}-4x-16-\left(x^{2}+26x+6\right)$$

To find the opposite of x^{2}+26x+6, find the opposite of each term.

$$4x^{2}-30x-34=2x^{2}-4x-16-x^{2}-26x-6$$

Combine 2x^{2} and -x^{2} to get x^{2}.

$$4x^{2}-30x-34=x^{2}-4x-16-26x-6$$

Combine -4x and -26x to get -30x.

$$4x^{2}-30x-34=x^{2}-30x-16-6$$

Subtract 6 from -16 to get -22.

$$4x^{2}-30x-34=x^{2}-30x-22$$

Subtract x^{2} from both sides.

$$4x^{2}-30x-34-x^{2}=-30x-22$$

Combine 4x^{2} and -x^{2} to get 3x^{2}.

$$3x^{2}-30x-34=-30x-22$$

Add 30x to both sides.

$$3x^{2}-30x-34+30x=-22$$

Combine -30x and 30x to get 0.

$$3x^{2}-34=-22$$

Add 34 to both sides.

$$3x^{2}=-22+34$$

Add -22 and 34 to get 12.

$$3x^{2}=12$$

Divide both sides by 3.

$$x^{2}=\frac{12}{3}$$

Divide 12 by 3 to get 4.

$$x^{2}=4$$

Take the square root of both sides of the equation.

$$x=2 x=-2$$

Variable x cannot be equal to -2.

$$x=2$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,4 since division by zero is not defined. Multiply both sides of the equation by \left(x-4\right)\left(x+2\right), the least common multiple of x+2,4-x,x^{2}-2x-8.

$$\left(x-4\right)\left(7x+7\right)-\left(2+x\right)\left(3x+3\right)=\left(x-4\right)\left(x+2\right)\times 2-\left(x^{2}+26x+6\right)$$

Use the distributive property to multiply x-4 by 7x+7 and combine like terms.

$$7x^{2}-21x-28-\left(2+x\right)\left(3x+3\right)=\left(x-4\right)\left(x+2\right)\times 2-\left(x^{2}+26x+6\right)$$

Use the distributive property to multiply -1 by 2+x.

$$7x^{2}-21x-28+\left(-2-x\right)\left(3x+3\right)=\left(x-4\right)\left(x+2\right)\times 2-\left(x^{2}+26x+6\right)$$

Use the distributive property to multiply -2-x by 3x+3 and combine like terms.

$$7x^{2}-21x-28-9x-6-3x^{2}=\left(x-4\right)\left(x+2\right)\times 2-\left(x^{2}+26x+6\right)$$

Combine -21x and -9x to get -30x.

$$7x^{2}-30x-28-6-3x^{2}=\left(x-4\right)\left(x+2\right)\times 2-\left(x^{2}+26x+6\right)$$

Subtract 6 from -28 to get -34.

$$7x^{2}-30x-34-3x^{2}=\left(x-4\right)\left(x+2\right)\times 2-\left(x^{2}+26x+6\right)$$

Combine 7x^{2} and -3x^{2} to get 4x^{2}.

$$4x^{2}-30x-34=\left(x-4\right)\left(x+2\right)\times 2-\left(x^{2}+26x+6\right)$$

Use the distributive property to multiply x-4 by x+2 and combine like terms.

$$4x^{2}-30x-34=\left(x^{2}-2x-8\right)\times 2-\left(x^{2}+26x+6\right)$$

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

$$4x^{2}-30x-34=2x^{2}-4x-16-\left(x^{2}+26x+6\right)$$

To find the opposite of x^{2}+26x+6, find the opposite of each term.

$$4x^{2}-30x-34=2x^{2}-4x-16-x^{2}-26x-6$$

Combine 2x^{2} and -x^{2} to get x^{2}.

$$4x^{2}-30x-34=x^{2}-4x-16-26x-6$$

Combine -4x and -26x to get -30x.

$$4x^{2}-30x-34=x^{2}-30x-16-6$$

Subtract 6 from -16 to get -22.

$$4x^{2}-30x-34=x^{2}-30x-22$$

Subtract x^{2} from both sides.

$$4x^{2}-30x-34-x^{2}=-30x-22$$

Combine 4x^{2} and -x^{2} to get 3x^{2}.

$$3x^{2}-30x-34=-30x-22$$

Add 30x to both sides.

$$3x^{2}-30x-34+30x=-22$$

Combine -30x and 30x to get 0.

$$3x^{2}-34=-22$$

Add 22 to both sides.

$$3x^{2}-34+22=0$$

Add -34 and 22 to get -12.

$$3x^{2}-12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 0 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 3\left(-12\right)}}{2\times 3}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 3\left(-12\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{0±\sqrt{-12\left(-12\right)}}{2\times 3}$$

Multiply -12 times -12.

$$x=\frac{0±\sqrt{144}}{2\times 3}$$

Take the square root of 144.

$$x=\frac{0±12}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{0±12}{6}$$

Now solve the equation x=\frac{0±12}{6} when ± is plus. Divide 12 by 6.

$$x=2$$

Now solve the equation x=\frac{0±12}{6} when ± is minus. Divide -12 by 6.

$$x=-2$$

The equation is now solved.

$$x=2 x=-2$$

Variable x cannot be equal to -2.

$$x=2$$