Solve (x-8)^2=0

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

Solve for x

$$x=8$$

Steps Using Factoring

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-8\right)^{2}.

$$x^{2}-16x+64=0$$

To solve the equation, factor x^{2}-16x+64 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.

$$a+b=-16 ab=64$$

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 64.

$$-1,-64 -2,-32 -4,-16 -8,-8$$

Calculate the sum for each pair.

$$-1-64=-65 -2-32=-34 -4-16=-20 -8-8=-16$$

The solution is the pair that gives sum -16.

$$a=-8 b=-8$$

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

$$\left(x-8\right)\left(x-8\right)$$

Rewrite as a binomial square.

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

To find equation solution, solve x-8=0.

$$x=8$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-8\right)^{2}.

$$x^{2}-16x+64=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+64. To find a and b, set up a system to be solved.

$$a+b=-16 ab=1\times 64=64$$

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 64.

$$-1,-64 -2,-32 -4,-16 -8,-8$$

Calculate the sum for each pair.

$$-1-64=-65 -2-32=-34 -4-16=-20 -8-8=-16$$

The solution is the pair that gives sum -16.

$$a=-8 b=-8$$

Rewrite x^{2}-16x+64 as \left(x^{2}-8x\right)+\left(-8x+64\right).

$$\left(x^{2}-8x\right)+\left(-8x+64\right)$$

Factor out x in the first and -8 in the second group.

$$x\left(x-8\right)-8\left(x-8\right)$$

Factor out common term x-8 by using distributive property.

$$\left(x-8\right)\left(x-8\right)$$

Rewrite as a binomial square.

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

To find equation solution, solve x-8=0.

$$x=8$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-8\right)^{2}.

$$x^{2}-16x+64=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -16 for b, and 64 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 64}}{2}$$

Square -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\times 64}}{2}$$

Multiply -4 times 64.

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

Add 256 to -256.

$$x=\frac{-\left(-16\right)±\sqrt{0}}{2}$$

Take the square root of 0.

$$x=-\frac{-16}{2}$$

The opposite of -16 is 16.

$$x=\frac{16}{2}$$

Divide 16 by 2.

$$x=8$$

Steps for Completing the Square

Take the square root of both sides of the equation.

$$\sqrt{\left(x-8\right)^{2}}=\sqrt{0}$$

Simplify.

$$x-8=0 x-8=0$$

Add 8 to both sides of the equation.

$$x=8 x=8$$

The equation is now solved. Solutions are the same.

$$x=8$$