Solve x+9/x+16x/x+9=8

$$\frac { x + 9 } { x } + \frac { 16 x } { x + 9 } = 8$$

Solve for x

$$x=3$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -9,0 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.

$$\left(x+9\right)\left(x+9\right)+x\times 16x=8x\left(x+9\right)$$

Multiply x+9 and x+9 to get \left(x+9\right)^{2}.

$$\left(x+9\right)^{2}+x\times 16x=8x\left(x+9\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+9\right)^{2}.

$$x^{2}+18x+81+x\times 16x=8x\left(x+9\right)$$

Multiply x and x to get x^{2}.

$$x^{2}+18x+81+x^{2}\times 16=8x\left(x+9\right)$$

Combine x^{2} and x^{2}\times 16 to get 17x^{2}.

$$17x^{2}+18x+81=8x\left(x+9\right)$$

Use the distributive property to multiply 8x by x+9.

$$17x^{2}+18x+81=8x^{2}+72x$$

Subtract 8x^{2} from both sides.

$$17x^{2}+18x+81-8x^{2}=72x$$

Combine 17x^{2} and -8x^{2} to get 9x^{2}.

$$9x^{2}+18x+81=72x$$

Subtract 72x from both sides.

$$9x^{2}+18x+81-72x=0$$

Combine 18x and -72x to get -54x.

$$9x^{2}-54x+81=0$$

Divide both sides by 9.

$$x^{2}-6x+9=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+9. To find a and b, set up a system to be solved.

$$a+b=-6 ab=1\times 9=9$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 9.

$$-1,-9 -3,-3$$

Calculate the sum for each pair.

$$-1-9=-10 -3-3=-6$$

The solution is the pair that gives sum -6.

$$a=-3 b=-3$$

Rewrite x^{2}-6x+9 as \left(x^{2}-3x\right)+\left(-3x+9\right).

$$\left(x^{2}-3x\right)+\left(-3x+9\right)$$

Factor out x in the first and -3 in the second group.

$$x\left(x-3\right)-3\left(x-3\right)$$

Factor out common term x-3 by using distributive property.

$$\left(x-3\right)\left(x-3\right)$$

Rewrite as a binomial square.

$$\left(x-3\right)^{2}$$

To find equation solution, solve x-3=0.

$$x=3$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -9,0 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.

$$\left(x+9\right)\left(x+9\right)+x\times 16x=8x\left(x+9\right)$$

Multiply x+9 and x+9 to get \left(x+9\right)^{2}.

$$\left(x+9\right)^{2}+x\times 16x=8x\left(x+9\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+9\right)^{2}.

$$x^{2}+18x+81+x\times 16x=8x\left(x+9\right)$$

Multiply x and x to get x^{2}.

$$x^{2}+18x+81+x^{2}\times 16=8x\left(x+9\right)$$

Combine x^{2} and x^{2}\times 16 to get 17x^{2}.

$$17x^{2}+18x+81=8x\left(x+9\right)$$

Use the distributive property to multiply 8x by x+9.

$$17x^{2}+18x+81=8x^{2}+72x$$

Subtract 8x^{2} from both sides.

$$17x^{2}+18x+81-8x^{2}=72x$$

Combine 17x^{2} and -8x^{2} to get 9x^{2}.

$$9x^{2}+18x+81=72x$$

Subtract 72x from both sides.

$$9x^{2}+18x+81-72x=0$$

Combine 18x and -72x to get -54x.

$$9x^{2}-54x+81=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, -54 for b, and 81 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-54\right)±\sqrt{\left(-54\right)^{2}-4\times 9\times 81}}{2\times 9}$$

Square -54.

$$x=\frac{-\left(-54\right)±\sqrt{2916-4\times 9\times 81}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-\left(-54\right)±\sqrt{2916-36\times 81}}{2\times 9}$$

Multiply -36 times 81.

$$x=\frac{-\left(-54\right)±\sqrt{2916-2916}}{2\times 9}$$

Add 2916 to -2916.

$$x=\frac{-\left(-54\right)±\sqrt{0}}{2\times 9}$$

Take the square root of 0.

$$x=-\frac{-54}{2\times 9}$$

The opposite of -54 is 54.

$$x=\frac{54}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{54}{18}$$

Divide 54 by 18.

$$x=3$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -9,0 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.

$$\left(x+9\right)\left(x+9\right)+x\times 16x=8x\left(x+9\right)$$

Multiply x+9 and x+9 to get \left(x+9\right)^{2}.

$$\left(x+9\right)^{2}+x\times 16x=8x\left(x+9\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+9\right)^{2}.

$$x^{2}+18x+81+x\times 16x=8x\left(x+9\right)$$

Multiply x and x to get x^{2}.

$$x^{2}+18x+81+x^{2}\times 16=8x\left(x+9\right)$$

Combine x^{2} and x^{2}\times 16 to get 17x^{2}.

$$17x^{2}+18x+81=8x\left(x+9\right)$$

Use the distributive property to multiply 8x by x+9.

$$17x^{2}+18x+81=8x^{2}+72x$$

Subtract 8x^{2} from both sides.

$$17x^{2}+18x+81-8x^{2}=72x$$

Combine 17x^{2} and -8x^{2} to get 9x^{2}.

$$9x^{2}+18x+81=72x$$

Subtract 72x from both sides.

$$9x^{2}+18x+81-72x=0$$

Combine 18x and -72x to get -54x.

$$9x^{2}-54x+81=0$$

Subtract 81 from both sides. Anything subtracted from zero gives its negation.

$$9x^{2}-54x=-81$$

Divide both sides by 9.

$$\frac{9x^{2}-54x}{9}=-\frac{81}{9}$$

Dividing by 9 undoes the multiplication by 9.

$$x^{2}+\left(-\frac{54}{9}\right)x=-\frac{81}{9}$$

Divide -54 by 9.

$$x^{2}-6x=-\frac{81}{9}$$

Divide -81 by 9.

$$x^{2}-6x=-9$$

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-6x+\left(-3\right)^{2}=-9+\left(-3\right)^{2}$$

Square -3.

$$x^{2}-6x+9=-9+9$$

Add -9 to 9.

$$x^{2}-6x+9=0$$

Factor x^{2}-6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-3\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{0}$$

Simplify.

$$x-3=0 x-3=0$$

Add 3 to both sides of the equation.

$$x=3 x=3$$

The equation is now solved. Solutions are the same.

$$x=3$$