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Chapter 6: Problem 6

Factor the difference of two squares. $$ 36 z^{2}-121 $$

Short Answer

Expert verified
The factored form of the difference of two squares \(36z^2 - 121\) is \((6z + 11)(6z - 11)\).

Step by step solution

01

Identify the Difference of Two Squares

In this exercise, the equation \(36z^2 - 121\) is presented. The two terms are \(36z^2\) and -121. Both these terms are perfect squares, therefore, the equation represents a difference of two squares.
02

Find the Square Root of Each Term

The square root of \(36z^2\) is \(6z\). The square root of 121 is 11. These two values will replace 'a' and 'b' in the equation \(a^2 - b^2 = (a+b) (a-b)\).
03

Apply the Difference of Squares Formula

To apply the formula, you need to add and subtract 'a' and 'b', which are \(6z\) and 11, respectively. This gives us: \((6z + 11)(6z - 11)\).

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Key Concepts

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

Perfect Squares
When we talk about perfect squares in algebra, we refer to expressions that are the result of squaring a number, a variable, or an algebraic expression. For example, the number 16 is a perfect square because it can be expressed as the square of 4, which is represented mathematically as \(4^2\text{ = 16}\). In similar fashion, \(36z^2\) from our exercise is a perfect square since it is equal to \(6z\) squared or \((6z)^2\).

Understanding perfect squares is crucial as they form the foundation for factoring differences of two squares. A strong grasp of perfect squares makes it easier to factor polynomial expressions and to simplify radical expressions as well.
Square Root
The square root of a number is a value that, when multiplied by itself, gives the original number. For every positive number 'a', there exists two square roots: a positive and a negative root, denoted as \(\sqrt{a}\) and \(-\sqrt{a}\), respectively. For example, the square root of 25 is 5, since \(5\times 5 = 25\). In algebra, finding the square root is essential when dealing with equations involving perfect squares.

In our exercise, we found the square roots of the terms \(36z^2\) and 121, which are \(6z\) and 11, respectively. Recognizing squares and their roots efficiently can significantly improve your ability to simplify expressions and solve equations.
Algebraic Expressions
An algebraic expression is a mathematical phrase that includes numbers, variables, and operation symbols. Examples include \(2x + 3\), \(4y^2 - 7y + 10\), and **our exercise's expression** \(36z^2 - 121\). Expressions differ from equations in that they do not contain an equals sign (\

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Most popular questions from this chapter

Solve the equation. $$ (x+10)(x+1)=36 $$

Factor the trinomial. $$ 6 m^{2}+7 m-20 $$

Evaluate the quantity mentally using the two samples as models. $$ \begin{array}{rlrl} 29^{2}=(30-1)^{2} & =30^{2}-2 \cdot 30 \cdot 1+1^{2} & 48 \cdot 52 & =(50-2)(50+2) \\ & =900-60+1=841 & & =50^{2}-2^{2}=2496 \end{array} $$ $$ 28 \cdot 32 $$

Estimate the \(x\)-intercepts of the graph of the equation. Set the quadratic equation equal to zero and solve. What do you notice? $$ y=2 x^{2}+x-3 $$

Solve the equation and check your solution.xplain how to identify and factor a perfect square trinomial. $$ 2-5(x-1)=2[x+10(x-1)] $$
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