Algebra Examples

$16 d^{2} - 24 d + 9$

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 $16 d^{2}$ . The square root of the first term is $\sqrt{16 d^{2}} = 4 d$ , so the first term is a perfect square.

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Step 2.1

Rewrite $16 d^{2}$ as ${(4 d)}^{2}$ .

$\sqrt{{(4 d)}^{2}}$

Step 2.2

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

$4 d$

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.

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Step 3.1

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

$\sqrt{3^{2}}$

Step 3.2

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

$3$

Step 4

The first term $16 d^{2}$ is a perfect square. The third term $9$ is a perfect square. The middle term $- 24 d$ is $- 2$ times the product of the square root of the first term $4 d$ and the square root of the third term $3$ .

The polynomial is a perfect square. ${(4 d - 3)}^{2}$