Solve (x-9)^2=0

$$( x - 9 ) ^ { 2 } = 0$$

Solve for x

$$x=9$$

Steps Using Factoring

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

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

To solve the equation, factor x^{2}-18x+81 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.

$$a+b=-18 ab=81$$

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 81.

$$-1,-81 -3,-27 -9,-9$$

Calculate the sum for each pair.

$$-1-81=-82 -3-27=-30 -9-9=-18$$

The solution is the pair that gives sum -18.

$$a=-9 b=-9$$

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

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

Rewrite as a binomial square.

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

To find equation solution, solve x-9=0.

$$x=9$$

Steps Using Factoring By Grouping

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

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+81. To find a and b, set up a system to be solved.

$$a+b=-18 ab=1\times 81=81$$

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 81.

$$-1,-81 -3,-27 -9,-9$$

Calculate the sum for each pair.

$$-1-81=-82 -3-27=-30 -9-9=-18$$

The solution is the pair that gives sum -18.

$$a=-9 b=-9$$

Rewrite x^{2}-18x+81 as \left(x^{2}-9x\right)+\left(-9x+81\right).

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

Factor out x in the first and -9 in the second group.

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

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

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

Rewrite as a binomial square.

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

To find equation solution, solve x-9=0.

$$x=9$$

Steps Using the Quadratic Formula

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

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -18 for b, and 81 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 81}}{2}$$

Square -18.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\times 81}}{2}$$

Multiply -4 times 81.

$$x=\frac{-\left(-18\right)±\sqrt{324-324}}{2}$$

Add 324 to -324.

$$x=\frac{-\left(-18\right)±\sqrt{0}}{2}$$

Take the square root of 0.

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

The opposite of -18 is 18.

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

Divide 18 by 2.

$$x=9$$

Steps for Completing the Square

Take the square root of both sides of the equation.

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

Simplify.

$$x-9=0 x-9=0$$

Add 9 to both sides of the equation.

$$x=9 x=9$$

The equation is now solved. Solutions are the same.

$$x=9$$