Solve 7/x+6/x-9=x-3/x-9

$$\frac { 7 } { x } + \frac { 6 } { x - 9 } = \frac { x - 3 } { x - 9 }$$

Solve for x

$$x=7$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values 0,9 since division by zero is not defined. Multiply both sides of the equation by x\left(x-9\right), the least common multiple of x,x-9.

$$\left(x-9\right)\times 7+x\times 6=x\left(x-3\right)$$

Use the distributive property to multiply x-9 by 7.

$$7x-63+x\times 6=x\left(x-3\right)$$

Combine 7x and x\times 6 to get 13x.

$$13x-63=x\left(x-3\right)$$

Use the distributive property to multiply x by x-3.

$$13x-63=x^{2}-3x$$

Subtract x^{2} from both sides.

$$13x-63-x^{2}=-3x$$

Add 3x to both sides.

$$13x-63-x^{2}+3x=0$$

Combine 13x and 3x to get 16x.

$$16x-63-x^{2}=0$$

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

$$-x^{2}+16x-63=0$$

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.

$$a+b=16 ab=-\left(-63\right)=63$$

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 63.

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

Calculate the sum for each pair.

$$1+63=64 3+21=24 7+9=16$$

The solution is the pair that gives sum 16.

$$a=9 b=7$$

Rewrite -x^{2}+16x-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 0,9 since division by zero is not defined. Multiply both sides of the equation by x\left(x-9\right), the least common multiple of x,x-9.

$$\left(x-9\right)\times 7+x\times 6=x\left(x-3\right)$$

Use the distributive property to multiply x-9 by 7.

$$7x-63+x\times 6=x\left(x-3\right)$$

Combine 7x and x\times 6 to get 13x.

$$13x-63=x\left(x-3\right)$$

Use the distributive property to multiply x by x-3.

$$13x-63=x^{2}-3x$$

Subtract x^{2} from both sides.

$$13x-63-x^{2}=-3x$$

Add 3x to both sides.

$$13x-63-x^{2}+3x=0$$

Combine 13x and 3x to get 16x.

$$16x-63-x^{2}=0$$

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.

$$-x^{2}+16x-63=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 16 for b, and -63 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\left(-1\right)\left(-63\right)}}{2\left(-1\right)}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\left(-1\right)\left(-63\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-16±\sqrt{256+4\left(-63\right)}}{2\left(-1\right)}$$

Multiply 4 times -63.

$$x=\frac{-16±\sqrt{256-252}}{2\left(-1\right)}$$

Add 256 to -252.

$$x=\frac{-16±\sqrt{4}}{2\left(-1\right)}$$

Take the square root of 4.

$$x=\frac{-16±2}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-16±2}{-2}$$

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

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

Divide -14 by -2.

$$x=7$$

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

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

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 0,9 since division by zero is not defined. Multiply both sides of the equation by x\left(x-9\right), the least common multiple of x,x-9.

$$\left(x-9\right)\times 7+x\times 6=x\left(x-3\right)$$

Use the distributive property to multiply x-9 by 7.

$$7x-63+x\times 6=x\left(x-3\right)$$

Combine 7x and x\times 6 to get 13x.

$$13x-63=x\left(x-3\right)$$

Use the distributive property to multiply x by x-3.

$$13x-63=x^{2}-3x$$

Subtract x^{2} from both sides.

$$13x-63-x^{2}=-3x$$

Add 3x to both sides.

$$13x-63-x^{2}+3x=0$$

Combine 13x and 3x to get 16x.

$$16x-63-x^{2}=0$$

Add 63 to both sides. Anything plus zero gives itself.

$$16x-x^{2}=63$$

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.

$$-x^{2}+16x=63$$

Divide both sides by -1.

$$\frac{-x^{2}+16x}{-1}=\frac{63}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{16}{-1}x=\frac{63}{-1}$$

Divide 16 by -1.

$$x^{2}-16x=\frac{63}{-1}$$

Divide 63 by -1.

$$x^{2}-16x=-63$$

Divide -16, the coefficient of the x term, by 2 to get -8. Then add the square of -8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-16x+\left(-8\right)^{2}=-63+\left(-8\right)^{2}$$

Square -8.

$$x^{2}-16x+64=-63+64$$

Add -63 to 64.

$$x^{2}-16x+64=1$$

Factor x^{2}-16x+64. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-8\right)^{2}=1$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-8\right)^{2}}=\sqrt{1}$$

Simplify.

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

Add 8 to both sides of the equation.

$$x=9 x=7$$

Variable x cannot be equal to 9.

$$x=7$$