Solve frac{(x-3)^{2}-(x+1)^{2}}{x^{2}-4x+3}-x+4/x^2-9=-17x+52-10x^2/x^3-x^2-9x+9

$$\frac { ( x - 3 ) ^ { 2 } - ( x + 1 ) ^ { 2 } } { x ^ { 2 } - 4 x + 3 } - \frac { x + 4 } { x ^ { 2 } - 9 } = \frac { - 17 x + 52 - 10 x ^ { 2 } } { x ^ { 3 } - x ^ { 2 } - 9 x + 9 }$$

Solve for x

$$x=-4$$

$$x=6$$

Steps Using Factoring

Variable x cannot be equal to any of the values -3,1,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x-1\right)\left(x+3\right), the least common multiple of x^{2}-4x+3,x^{2}-9,x^{3}-x^{2}-9x+9.

$$\left(x+3\right)\left(\left(x-3\right)^{2}-\left(x+1\right)^{2}\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$\left(x+3\right)\left(x^{2}-6x+9-\left(x+1\right)^{2}\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$\left(x+3\right)\left(x^{2}-6x+9-\left(x^{2}+2x+1\right)\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}$$

To find the opposite of x^{2}+2x+1, find the opposite of each term.

$$\left(x+3\right)\left(x^{2}-6x+9-x^{2}-2x-1\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}$$

Combine x^{2} and -x^{2} to get 0.

$$\left(x+3\right)\left(-6x+9-2x-1\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}$$

Combine -6x and -2x to get -8x.

$$\left(x+3\right)\left(-8x+9-1\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}$$

Subtract 1 from 9 to get 8.

$$\left(x+3\right)\left(-8x+8\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}$$

Use the distributive property to multiply x+3 by -8x+8 and combine like terms.

$$-8x^{2}-16x+24-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}$$

Use the distributive property to multiply x-1 by x+4 and combine like terms.

$$-8x^{2}-16x+24-\left(x^{2}+3x-4\right)=-17x+52-10x^{2}$$

To find the opposite of x^{2}+3x-4, find the opposite of each term.

$$-8x^{2}-16x+24-x^{2}-3x+4=-17x+52-10x^{2}$$

Combine -8x^{2} and -x^{2} to get -9x^{2}.

$$-9x^{2}-16x+24-3x+4=-17x+52-10x^{2}$$

Combine -16x and -3x to get -19x.

$$-9x^{2}-19x+24+4=-17x+52-10x^{2}$$

Add 24 and 4 to get 28.

$$-9x^{2}-19x+28=-17x+52-10x^{2}$$

Add 17x to both sides.

$$-9x^{2}-19x+28+17x=52-10x^{2}$$

Combine -19x and 17x to get -2x.

$$-9x^{2}-2x+28=52-10x^{2}$$

Subtract 52 from both sides.

$$-9x^{2}-2x+28-52=-10x^{2}$$

Subtract 52 from 28 to get -24.

$$-9x^{2}-2x-24=-10x^{2}$$

Add 10x^{2} to both sides.

$$-9x^{2}-2x-24+10x^{2}=0$$

Combine -9x^{2} and 10x^{2} to get x^{2}.

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

To solve the equation, factor x^{2}-2x-24 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-2 ab=-24$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -24.

$$1,-24 2,-12 3,-8 4,-6$$

Calculate the sum for each pair.

$$1-24=-23 2-12=-10 3-8=-5 4-6=-2$$

The solution is the pair that gives sum -2.

$$a=-6 b=4$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-6\right)\left(x+4\right)$$

To find equation solutions, solve x-6=0 and x+4=0.

$$x=6 x=-4$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -3,1,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x-1\right)\left(x+3\right), the least common multiple of x^{2}-4x+3,x^{2}-9,x^{3}-x^{2}-9x+9.

$$\left(x+3\right)\left(\left(x-3\right)^{2}-\left(x+1\right)^{2}\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$\left(x+3\right)\left(x^{2}-6x+9-\left(x+1\right)^{2}\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$\left(x+3\right)\left(x^{2}-6x+9-\left(x^{2}+2x+1\right)\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}$$

To find the opposite of x^{2}+2x+1, find the opposite of each term.

$$\left(x+3\right)\left(x^{2}-6x+9-x^{2}-2x-1\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}$$

Combine x^{2} and -x^{2} to get 0.

$$\left(x+3\right)\left(-6x+9-2x-1\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}$$

Combine -6x and -2x to get -8x.

$$\left(x+3\right)\left(-8x+9-1\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}$$

Subtract 1 from 9 to get 8.

$$\left(x+3\right)\left(-8x+8\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}$$

Use the distributive property to multiply x+3 by -8x+8 and combine like terms.

$$-8x^{2}-16x+24-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}$$

Use the distributive property to multiply x-1 by x+4 and combine like terms.

$$-8x^{2}-16x+24-\left(x^{2}+3x-4\right)=-17x+52-10x^{2}$$

To find the opposite of x^{2}+3x-4, find the opposite of each term.

$$-8x^{2}-16x+24-x^{2}-3x+4=-17x+52-10x^{2}$$

Combine -8x^{2} and -x^{2} to get -9x^{2}.

$$-9x^{2}-16x+24-3x+4=-17x+52-10x^{2}$$

Combine -16x and -3x to get -19x.

$$-9x^{2}-19x+24+4=-17x+52-10x^{2}$$

Add 24 and 4 to get 28.

$$-9x^{2}-19x+28=-17x+52-10x^{2}$$

Add 17x to both sides.

$$-9x^{2}-19x+28+17x=52-10x^{2}$$

Combine -19x and 17x to get -2x.

$$-9x^{2}-2x+28=52-10x^{2}$$

Subtract 52 from both sides.

$$-9x^{2}-2x+28-52=-10x^{2}$$

Subtract 52 from 28 to get -24.

$$-9x^{2}-2x-24=-10x^{2}$$

Add 10x^{2} to both sides.

$$-9x^{2}-2x-24+10x^{2}=0$$

Combine -9x^{2} and 10x^{2} to get x^{2}.

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-24. To find a and b, set up a system to be solved.

$$a+b=-2 ab=1\left(-24\right)=-24$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -24.

$$1,-24 2,-12 3,-8 4,-6$$

Calculate the sum for each pair.

$$1-24=-23 2-12=-10 3-8=-5 4-6=-2$$

The solution is the pair that gives sum -2.

$$a=-6 b=4$$

Rewrite x^{2}-2x-24 as \left(x^{2}-6x\right)+\left(4x-24\right).

$$\left(x^{2}-6x\right)+\left(4x-24\right)$$

Factor out x in the first and 4 in the second group.

$$x\left(x-6\right)+4\left(x-6\right)$$

Factor out common term x-6 by using distributive property.

$$\left(x-6\right)\left(x+4\right)$$

To find equation solutions, solve x-6=0 and x+4=0.

$$x=6 x=-4$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -3,1,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x-1\right)\left(x+3\right), the least common multiple of x^{2}-4x+3,x^{2}-9,x^{3}-x^{2}-9x+9.

$$\left(x+3\right)\left(\left(x-3\right)^{2}-\left(x+1\right)^{2}\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$\left(x+3\right)\left(x^{2}-6x+9-\left(x+1\right)^{2}\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$\left(x+3\right)\left(x^{2}-6x+9-\left(x^{2}+2x+1\right)\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}$$

To find the opposite of x^{2}+2x+1, find the opposite of each term.

$$\left(x+3\right)\left(x^{2}-6x+9-x^{2}-2x-1\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}$$

Combine x^{2} and -x^{2} to get 0.

$$\left(x+3\right)\left(-6x+9-2x-1\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}$$

Combine -6x and -2x to get -8x.

$$\left(x+3\right)\left(-8x+9-1\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}$$

Subtract 1 from 9 to get 8.

$$\left(x+3\right)\left(-8x+8\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}$$

Use the distributive property to multiply x+3 by -8x+8 and combine like terms.

$$-8x^{2}-16x+24-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}$$

Use the distributive property to multiply x-1 by x+4 and combine like terms.

$$-8x^{2}-16x+24-\left(x^{2}+3x-4\right)=-17x+52-10x^{2}$$

To find the opposite of x^{2}+3x-4, find the opposite of each term.

$$-8x^{2}-16x+24-x^{2}-3x+4=-17x+52-10x^{2}$$

Combine -8x^{2} and -x^{2} to get -9x^{2}.

$$-9x^{2}-16x+24-3x+4=-17x+52-10x^{2}$$

Combine -16x and -3x to get -19x.

$$-9x^{2}-19x+24+4=-17x+52-10x^{2}$$

Add 24 and 4 to get 28.

$$-9x^{2}-19x+28=-17x+52-10x^{2}$$

Add 17x to both sides.

$$-9x^{2}-19x+28+17x=52-10x^{2}$$

Combine -19x and 17x to get -2x.

$$-9x^{2}-2x+28=52-10x^{2}$$

Subtract 52 from both sides.

$$-9x^{2}-2x+28-52=-10x^{2}$$

Subtract 52 from 28 to get -24.

$$-9x^{2}-2x-24=-10x^{2}$$

Add 10x^{2} to both sides.

$$-9x^{2}-2x-24+10x^{2}=0$$

Combine -9x^{2} and 10x^{2} to get x^{2}.

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

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

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

Square -2.

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

Multiply -4 times -24.

$$x=\frac{-\left(-2\right)±\sqrt{4+96}}{2}$$

Add 4 to 96.

$$x=\frac{-\left(-2\right)±\sqrt{100}}{2}$$

Take the square root of 100.

$$x=\frac{-\left(-2\right)±10}{2}$$

The opposite of -2 is 2.

$$x=\frac{2±10}{2}$$

Now solve the equation x=\frac{2±10}{2} when ± is plus. Add 2 to 10.

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

Divide 12 by 2.

$$x=6$$

Now solve the equation x=\frac{2±10}{2} when ± is minus. Subtract 10 from 2.

$$x=-\frac{8}{2}$$

Divide -8 by 2.

$$x=-4$$

The equation is now solved.

$$x=6 x=-4$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -3,1,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x-1\right)\left(x+3\right), the least common multiple of x^{2}-4x+3,x^{2}-9,x^{3}-x^{2}-9x+9.

$$\left(x+3\right)\left(\left(x-3\right)^{2}-\left(x+1\right)^{2}\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$\left(x+3\right)\left(x^{2}-6x+9-\left(x+1\right)^{2}\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$\left(x+3\right)\left(x^{2}-6x+9-\left(x^{2}+2x+1\right)\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}$$

To find the opposite of x^{2}+2x+1, find the opposite of each term.

$$\left(x+3\right)\left(x^{2}-6x+9-x^{2}-2x-1\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}$$

Combine x^{2} and -x^{2} to get 0.

$$\left(x+3\right)\left(-6x+9-2x-1\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}$$

Combine -6x and -2x to get -8x.

$$\left(x+3\right)\left(-8x+9-1\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}$$

Subtract 1 from 9 to get 8.

$$\left(x+3\right)\left(-8x+8\right)-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}$$

Use the distributive property to multiply x+3 by -8x+8 and combine like terms.

$$-8x^{2}-16x+24-\left(x-1\right)\left(x+4\right)=-17x+52-10x^{2}$$

Use the distributive property to multiply x-1 by x+4 and combine like terms.

$$-8x^{2}-16x+24-\left(x^{2}+3x-4\right)=-17x+52-10x^{2}$$

To find the opposite of x^{2}+3x-4, find the opposite of each term.

$$-8x^{2}-16x+24-x^{2}-3x+4=-17x+52-10x^{2}$$

Combine -8x^{2} and -x^{2} to get -9x^{2}.

$$-9x^{2}-16x+24-3x+4=-17x+52-10x^{2}$$

Combine -16x and -3x to get -19x.

$$-9x^{2}-19x+24+4=-17x+52-10x^{2}$$

Add 24 and 4 to get 28.

$$-9x^{2}-19x+28=-17x+52-10x^{2}$$

Add 17x to both sides.

$$-9x^{2}-19x+28+17x=52-10x^{2}$$

Combine -19x and 17x to get -2x.

$$-9x^{2}-2x+28=52-10x^{2}$$

Add 10x^{2} to both sides.

$$-9x^{2}-2x+28+10x^{2}=52$$

Combine -9x^{2} and 10x^{2} to get x^{2}.

$$x^{2}-2x+28=52$$

Subtract 28 from both sides.

$$x^{2}-2x=52-28$$

Subtract 28 from 52 to get 24.

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

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2x+1=24+1$$

Add 24 to 1.

$$x^{2}-2x+1=25$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{25}$$

Simplify.

$$x-1=5 x-1=-5$$

Add 1 to both sides of the equation.

$$x=6 x=-4$$