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Q12.

Expert-verified

Algebra 1

Found in: Page 523

Book edition Student Edition

Author(s) Carter, Cuevas, Day, Holiday, Luchin

Pages 801 pages

ISBN 9780078884801

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Short Answer

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

Yes, the given trinomial is a perfect square trinomial.

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

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

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

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

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

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

$4 x^{2} = {(2 x)}^{2}$

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

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

$49 = {(7)}^{2}$

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

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

$28 x = 2 (2 x) (7)$

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:

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

It can be noticed that:

$4 x^{2} + 28 x + 49 = {(2 x)}^{2} + 2 (2 x) (7) + {(7)}^{2} = (2 x + 7) (2 x + 7) (∵ a^{2} + 2 a b + b^{2} = (a + b) (a + b) = {(a + b)}^{2}) = {(2 x + 7)}^{2}$

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

Most popular questions for Math Textbooks

Write an equation in slope-intercept form with a slope of $\frac{9}{10}$ and y-intercept of 3.

$F y = 3 x + \frac{9}{10} G y = \frac{9}{10} x + 3 H y = \frac{9}{10} x - 3 J y = 3 x - \frac{9}{10}$

Solve the equation by using the Quadratic formula. Round to the nearest tenth if necessary.

$x^{2} - 10 x = - 15$

Graph each function. Find the $y$ -intercept, and state the domain and range.

$y = 3^{x} + 1$

Describe how the graph of the function $f (x) = x^{2} - 3$ is related to the graph $f (x) = x^{2}$ .

Graph each function. Find the y-intercept, and state the domain and range.

$y = 3 x + 2$

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