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Chapter 6: Problem 18

Factor completely. $$ a^{2}+16 a+64 $$

Short Answer

Expert verified
(a + 8)^2

Step by step solution

01

Identify the structure of the quadratic equation

The given equation is in the standard quadratic form: \[a^2 + 16a + 64\]. This can be factored as a perfect square trinomial.
02

Identify the perfect square trinomial

A perfect square trinomial can be written as: \[x^2 + 2xy + y^2 = (x + y)^2\]. Compare this with our equation: \[a^2 + 16a + 64\]. We see that \(x = a\) and \(y = 8\), since \(2xy = 16a\) and \(8^2 = 64\).
03

Write the factored form

Using the identified values, the equation can be rewritten as: \[a^2 + 16a + 64 = (a + 8)^2\].
04

State the final factored form

The completely factored form of \[a^2 + 16a + 64\] is \[(a + 8)^2\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Perfect Square Trinomial
In algebra, a perfect square trinomial is a very special kind of polynomial that follows a specific pattern. It looks like this: To recognize a perfect square trinomial, check if the first and last terms are perfect squares. Also, see if the middle term is twice the product of the roots of these squares. For instance: Pet, cite this example: which simplifies to We can then express it as In general, a trinomial in the form Expanding the square will yield Therefore, the given equation The method of identifying These trinomials simplify beautifully into a squared binomial, making factoring easy and straightforward.
Quadratic Equation
A quadratic equation is a second-degree polynomial typically written in the form Here, 'a,' 'b,' and 'c'quadratic equations are To identify the correct form Here, we can rewrite it. The roots of In our problem, Notice the terms Quadratic equations are fundamental in algebra. They recur across various mathematical problems and real-world scenarios, like motion and area calculations.
Factored Form
Factoring transforms a polynomial into a product of simpler polynomials. For quadratics, factored form makes equations easier to solve and understand. The factored form is typically written as The benefit of factoring For the given quadratic This makes calculations simpler Factored forms significantly simplify finding solutions to equations. They show the zeros or roots explicitly, making graphing and problem-solving effortless. It's a crucial skill in algebra that applies to higher-level math and various scientific fields.

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Most popular questions from this chapter

Solve. Round any irrational solutions to the nearest thousandth. $$ x^{2}=\frac{1}{100} $$

Determine whether each of the following is a difference of squares. $$ -12+25 t^{2} $$

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Ashlee factors \(8 x^{2} y-10 x y^{2}\) as $$ 2 x y \cdot 4 x-2 x y \cdot 5 y $$ Is this the factorization of the polynomial? Why or why not?

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ 9 a^{4}-25 a^{2} b^{4} $$
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