Solve x^2+2*x*9+9^2

$${ x }^{ 2 } +2 \times x \times 9+ { 9 }^{ 2 }$$

Evaluate

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

Solution Steps

Multiply 2 and 9 to get 18.

$$x^{2}+18x+9^{2}$$

Calculate 9 to the power of 2 and get 81.

$$x^{2}+18x+81$$

Factor

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

Solution Steps

Multiply and combine like terms.

$$x^{2}+18x+81$$

Factor the expression by grouping. First, the expression 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 positive, a and b are both positive. 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.

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