Question

Which trinomials are perfect square trinomials?

Select each correct answer.

A. m^2 + 10m + 100

B. m^2 + 10m + 25

C. m^2 + 36m + 60

D. m^2 + 24m + 144

Answer 1

ANSWER

B. ${m}^{2}+10m+25$

D. ${m}^{2} + 24m+144$

EXPLANATION

A perfect square trinomial is of the form;

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

or

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

For option A,

${m}^{2} + 2(5)m + {10}^{2} \ne \: {(m + 10)}^{2}$

This is not a perfect square trinomial.

For option B

${m}^{2} + 2(5)m + {5}^{2} = \: {(m + 5)}^{2}$

This is a perfect square trinomial.

For option C

${m}^{2} + 2(18)m + {6}^{2} \ne \: {(m + 6)}^{2} .$

This is not a perfect square trinomial.

For option D.

${m}^{2} + 2(12)m + {12}^{2} = \: {(m + 12)}^{2}$

This is a perfect square trinomial.

The correct answers are option B and D.

Answer 2

You can use the definition of perfect square trinomials to find which of the given trinomials are perfect squares.

The trinomials which are perfect squares are:

Option B: $m^2 + 10m + 25$

Option D: $m^2 + 24m + 144$

What are perfect squares trinomials?

They are those expressions which are found by squaring binomial expressions.

How to find which of the given trinomials are perfect square trinomials?

Since the given trinomials are with degree 2, thus, if they are perfect square, the binomial which was used to make them must be linear.

Let the binomial term was am + b(a linear expression is always writable in this form where a and b are constants and m is a variable), then we will obtain:

$(am +b )^2 = a^2m^2 + b^2 + 2abm$

Checking all options:

• Option A: $m^2 + 10m + 100$

Comparing it with the standard form we obtained, we have:

$a = \pm\sqrt{1} = \pm1\\b = \pm\sqrt{100} = \pm10\\\\2ab = \pm20\\10 = \pm 20\\\text{since above statement is wrong, thus this option is not a perfect square trinomial}$

• Option B: $m^2 + 10m + 25$

Comparing it with the standard form we obtained, we have:

$a = \pm\sqrt{1} = \pm1\\b = \pm\sqrt{25} = \pm5\\\\2ab = \pm10\\10 = \pm 10\\\text{since above statement is correct for + sign, thus we have:}\\m^2 + 10m + 25 = (m + 5)^2$

Thus, this trinomial is perfect squares trinomial.

• Option C: $m^2 + 36m + 60$

Comparing it with the standard form we obtained, we have:

$a = \pm\sqrt{1} = \pm1\\b = \pm\sqrt{60} = \pm2\sqrt{15}\\\\2ab = \pm4\sqrt{15}\\36 = \pm 4\sqrt{15}\\\text{since above statement is wrong, thus this option is not a perfect square trinomial}$

Option D: $m^2 + 24m + 144$

Comparing it with the standard form we obtained, we have:

$a = \pm\sqrt{1} = \pm1\\b = \pm\sqrt{144} = \pm12\\\\2ab = \pm24\\24 = \pm 20\\\text{since above statement is correct for + sign, thus we have:}\\\\m^2 + 24m + 144 = (m+12)^2$

Thus,   this trinomial is perfect squares trinomial.

Thus,

The trinomials which are perfect squares are:

Option B: $m^2 + 10m + 25$

Option D: $m^2 + 24m + 144$

Learn more about perfect square trinomials here:

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