Solve x-9/x^2-12x+27-x+8/x-3=0

$$\frac { x - 9 } { x ^ { 2 } - 12 x + 27 } - \frac { x + 8 } { x - 3 } = 0$$

Solve for x

$$x=-7$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values 3,9 since division by zero is not defined. Multiply both sides of the equation by \left(x-9\right)\left(x-3\right), the least common multiple of x^{2}-12x+27,x-3.

$$x-9-\left(x-9\right)\left(x+8\right)=0$$

Use the distributive property to multiply x-9 by x+8 and combine like terms.

$$x-9-\left(x^{2}-x-72\right)=0$$

To find the opposite of x^{2}-x-72, find the opposite of each term.

$$x-9-x^{2}+x+72=0$$

Combine x and x to get 2x.

$$2x-9-x^{2}+72=0$$

Add -9 and 72 to get 63.

$$2x+63-x^{2}=0$$

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

$$-x^{2}+2x+63=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+63. To find a and b, set up a system to be solved.

$$a+b=2 ab=-63=-63$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -63.

$$-1,63 -3,21 -7,9$$

Calculate the sum for each pair.

$$-1+63=62 -3+21=18 -7+9=2$$

The solution is the pair that gives sum 2.

$$a=9 b=-7$$

Rewrite -x^{2}+2x+63 as \left(-x^{2}+9x\right)+\left(-7x+63\right).

$$\left(-x^{2}+9x\right)+\left(-7x+63\right)$$

Factor out -x in the first and -7 in the second group.

$$-x\left(x-9\right)-7\left(x-9\right)$$

Factor out common term x-9 by using distributive property.

$$\left(x-9\right)\left(-x-7\right)$$

To find equation solutions, solve x-9=0 and -x-7=0.

$$x=9 x=-7$$

Variable x cannot be equal to 9.

$$x=-7$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 3,9 since division by zero is not defined. Multiply both sides of the equation by \left(x-9\right)\left(x-3\right), the least common multiple of x^{2}-12x+27,x-3.

$$x-9-\left(x-9\right)\left(x+8\right)=0$$

Use the distributive property to multiply x-9 by x+8 and combine like terms.

$$x-9-\left(x^{2}-x-72\right)=0$$

To find the opposite of x^{2}-x-72, find the opposite of each term.

$$x-9-x^{2}+x+72=0$$

Combine x and x to get 2x.

$$2x-9-x^{2}+72=0$$

Add -9 and 72 to get 63.

$$2x+63-x^{2}=0$$

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.

$$-x^{2}+2x+63=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 2 for b, and 63 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-1\right)\times 63}}{2\left(-1\right)}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-1\right)\times 63}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-2±\sqrt{4+4\times 63}}{2\left(-1\right)}$$

Multiply 4 times 63.

$$x=\frac{-2±\sqrt{4+252}}{2\left(-1\right)}$$

Add 4 to 252.

$$x=\frac{-2±\sqrt{256}}{2\left(-1\right)}$$

Take the square root of 256.

$$x=\frac{-2±16}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-2±16}{-2}$$

Now solve the equation x=\frac{-2±16}{-2} when ± is plus. Add -2 to 16.

$$x=\frac{14}{-2}$$

Divide 14 by -2.

$$x=-7$$

Now solve the equation x=\frac{-2±16}{-2} when ± is minus. Subtract 16 from -2.

$$x=-\frac{18}{-2}$$

Divide -18 by -2.

$$x=9$$

The equation is now solved.

$$x=-7 x=9$$

Variable x cannot be equal to 9.

$$x=-7$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 3,9 since division by zero is not defined. Multiply both sides of the equation by \left(x-9\right)\left(x-3\right), the least common multiple of x^{2}-12x+27,x-3.

$$x-9-\left(x-9\right)\left(x+8\right)=0$$

Use the distributive property to multiply x-9 by x+8 and combine like terms.

$$x-9-\left(x^{2}-x-72\right)=0$$

To find the opposite of x^{2}-x-72, find the opposite of each term.

$$x-9-x^{2}+x+72=0$$

Combine x and x to get 2x.

$$2x-9-x^{2}+72=0$$

Add -9 and 72 to get 63.

$$2x+63-x^{2}=0$$

Subtract 63 from both sides. Anything subtracted from zero gives its negation.

$$2x-x^{2}=-63$$

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.

$$-x^{2}+2x=-63$$

Divide both sides by -1.

$$\frac{-x^{2}+2x}{-1}=-\frac{63}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{2}{-1}x=-\frac{63}{-1}$$

Divide 2 by -1.

$$x^{2}-2x=-\frac{63}{-1}$$

Divide -63 by -1.

$$x^{2}-2x=63$$

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.

$$x^{2}-2x+1=63+1$$

Add 63 to 1.

$$x^{2}-2x+1=64$$

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}.

$$\left(x-1\right)^{2}=64$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{64}$$

Simplify.

$$x-1=8 x-1=-8$$

Add 1 to both sides of the equation.

$$x=9 x=-7$$

Variable x cannot be equal to 9.

$$x=-7$$