Determine if the Expression is a Perfect Square 9x^2-18x+9

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

Step 1

A trinomial can be a perfect square if it satisfies the following:

The first term is a perfect square.

The third term is a perfect square.

The middle term is either $2$ or $- 2$ times the product of the square root of the first term and the square root of the third term.

${(a - b)}^{2} = a^{2} - 2 a b + b^{2}$

Step 2

Find $a$ , which is the square root of the first term $9 x^{2}$ . The square root of the first term is $\sqrt{9 x^{2}} = 3 x$ , so the first term is a perfect square.

Rewrite $9 x^{2}$ as ${(3 x)}^{2}$ .

$\sqrt{{(3 x)}^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$3 x$

Step 3

Find $b$ , which is the square root of the third term $9$ . The square root of the third term is $\sqrt{9} = 3$ , so the third term is a perfect square.

Rewrite $9$ as $3^{2}$ .

$\sqrt{3^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$3$

Step 4

The first term $9 x^{2}$ is a perfect square. The third term $9$ is a perfect square. The middle term $- 18 x$ is $- 2$ times the product of the square root of the first term $3 x$ and the square root of the third term $3$ .

The polynomial is a perfect square. $9 {(x - 1)}^{2}$

Determine if the Expression is a Perfect Square 9x^2-18x+9

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