Solve (x-15)^2/8-x=0

$$\frac { ( x - 15 ) ^ { 2 } } { 8 - x } = 0$$

Solve for x

$$x=15$$

Steps Using Factoring

Variable x cannot be equal to 8 since division by zero is not defined. Multiply both sides of the equation by -x+8.

$$\left(x-15\right)^{2}=0$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-15\right)^{2}.

$$x^{2}-30x+225=0$$

To solve the equation, factor x^{2}-30x+225 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.

$$a+b=-30 ab=225$$

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 225.

$$-1,-225 -3,-75 -5,-45 -9,-25 -15,-15$$

Calculate the sum for each pair.

$$-1-225=-226 -3-75=-78 -5-45=-50 -9-25=-34 -15-15=-30$$

The solution is the pair that gives sum -30.

$$a=-15 b=-15$$

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

$$\left(x-15\right)\left(x-15\right)$$

Rewrite as a binomial square.

$$\left(x-15\right)^{2}$$

To find equation solution, solve x-15=0.

$$x=15$$

Steps Using Factoring By Grouping

Variable x cannot be equal to 8 since division by zero is not defined. Multiply both sides of the equation by -x+8.

$$\left(x-15\right)^{2}=0$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-15\right)^{2}.

$$x^{2}-30x+225=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+225. To find a and b, set up a system to be solved.

$$a+b=-30 ab=1\times 225=225$$

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 225.

$$-1,-225 -3,-75 -5,-45 -9,-25 -15,-15$$

Calculate the sum for each pair.

$$-1-225=-226 -3-75=-78 -5-45=-50 -9-25=-34 -15-15=-30$$

The solution is the pair that gives sum -30.

$$a=-15 b=-15$$

Rewrite x^{2}-30x+225 as \left(x^{2}-15x\right)+\left(-15x+225\right).

$$\left(x^{2}-15x\right)+\left(-15x+225\right)$$

Factor out x in the first and -15 in the second group.

$$x\left(x-15\right)-15\left(x-15\right)$$

Factor out common term x-15 by using distributive property.

$$\left(x-15\right)\left(x-15\right)$$

Rewrite as a binomial square.

$$\left(x-15\right)^{2}$$

To find equation solution, solve x-15=0.

$$x=15$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 8 since division by zero is not defined. Multiply both sides of the equation by -x+8.

$$\left(x-15\right)^{2}=0$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-15\right)^{2}.

$$x^{2}-30x+225=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -30 for b, and 225 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}-4\times 225}}{2}$$

Square -30.

$$x=\frac{-\left(-30\right)±\sqrt{900-4\times 225}}{2}$$

Multiply -4 times 225.

$$x=\frac{-\left(-30\right)±\sqrt{900-900}}{2}$$

Add 900 to -900.

$$x=\frac{-\left(-30\right)±\sqrt{0}}{2}$$

Take the square root of 0.

$$x=-\frac{-30}{2}$$

The opposite of -30 is 30.

$$x=\frac{30}{2}$$

Divide 30 by 2.

$$x=15$$

Steps for Completing the Square

Variable x cannot be equal to 8 since division by zero is not defined. Multiply both sides of the equation by -x+8.

$$\left(x-15\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-15\right)^{2}}=\sqrt{0}$$

Simplify.

$$x-15=0 x-15=0$$

Add 15 to both sides of the equation.

$$x=15 x=15$$

The equation is now solved. Solutions are the same.

$$x=15$$