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Q12.

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Found in: Page 523

### Algebra 1

Book edition Student Edition
Author(s) Carter, Cuevas, Day, Holiday, Luchin
Pages 801 pages
ISBN 9780078884801

# Determine whether each trinomial is a perfect square trinomial. Write yes or no. If so, factor it.$4{x}^{2}+28x+49$

Yes, the given trinomial is a perfect square trinomial.

The factorization of the given trinomial is ${\left(2x+7\right)}^{\mathbf{2}}$.

See the step by step solution

## Step 1. Observe the given trinomial 4x2+28x+49.

The given trinomial is: $4{x}^{2}+28x+49$

The First, middle and last terms of the given trinomial are $4{x}^{2},28x$ and 49 respectively.

The first term of the given trinomial can be written as:

$4{x}^{2}={\left(2x\right)}^{2}$

Therefore, the first term of the given trinomial is a perfect square.

The last term of the given trinomial can be written as:

$49={\left(7\right)}^{2}$

Therefore, the last term of the given trinomial is a perfect square.

The middle term of the given trinomial can be written as:

$28x=2\left(2x\right)\left(7\right)$

Therefore, the middle term is twice the product of the square roots of the first term and last term.

## Step 2. Determine whether the given trinomial 4x2+28x+49 is a perfect square trinomial.

As the first and last terms of the given trinomial are a perfect square and the middle term is twice the product of the square roots of the first term and last term.

Therefore, yes, the given trinomial is a perfect square trinomial.

## Step 3. Factor the given trinomial 4x2+28x+49.

It is known that:

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

It can be noticed that:

$4{x}^{2}+28x+49={\left(2x\right)}^{2}+2\left(2x\right)\left(7\right)+{\left(7\right)}^{2}\phantom{\rule{0ex}{0ex}}=\left(2x+7\right)\left(2x+7\right)\text{}\left(\because \text{}{a}^{2}+2ab+{b}^{2}=\left(a+b\right)\left(a+b\right)={\left(a+b\right)}^{2}\right)\phantom{\rule{0ex}{0ex}}={\left(2x+7\right)}^{2}$

Therefore, the factorization of the given trinomial is ${\left(2x+7\right)}^{2}$.