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Chapter 10: Problem 19

Factor the expression. $$ 100 x^{2}-121 $$

Short Answer

Expert verified
The factored expression is \((10x - 11)(10x + 11)\).

Step by step solution

01

Identify the square terms

Looking at our expression \(100x^2 - 121\), we can see that \(100x^2 = (10x)^2\) and \(121 = 11^2\). So our \(a\) is \(10x\) and our \(b\) is \(11\).
02

Apply the difference of squares formula

A difference of squares is a special algebraic identity in the excerpted form \(a^2 - b^2\), which can be factored into \((a-b)(a+b)\). By substituting the identified \(a\) and \(b\) from step 1 into the formula, the factored form of the expression becomes \((10x - 11)(10x + 11)\).

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Key Concepts

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

Factoring
Factoring is a key skill in algebra that makes complex expressions more manageable. In our exercise, the aim is to factor the expression \(100x^2 - 121\). To factor means to break down an expression into parts that when multiplied together give back the original expression.
In this case, you need to look for patterns, like the difference of squares, that can simplify the factoring process. Factoring helps in solving equations easily and finding roots without hassle.
  • First, identify if there is a common factor among the terms. For example, in basic forms of factoring, you might look to take out common numbers or variables.

  • Next, apply specific formulas or identities, such as the difference of squares, to rewrite the expression neatly. This process not only simplifies the expression but also helps in visualizing the solution.

Understanding how to factor correctly is crucial for solving algebraic equations seamlessly.
Algebraic Identities
Algebraic identities are like shortcuts that help in simplifying and solving algebraic expressions quickly. They are proven formulas that work for any numbers or variables included in them. In the exercise provided, the key algebraic identity used was the difference of squares.
This particular algebraic identity states that \(a^2 - b^2 = (a-b)(a+b)\). It allows you to break down the expression into two binomials effortlessly.
  • The identity is derived from multiplying two conjugates, such as \((a - b)\) and \((a + b)\), and observing that the middle terms cancel each other out.

  • Using identities like these saves time and minimizes errors, especially when dealing with more complicated quadratic expressions.

Developing a strong understanding of algebraic identities supports problem-solving in algebra and other areas of mathematics.
Quadratic Expressions
Quadratic expressions are polynomials of degree two, and they often appear in the form \(ax^2 + bx + c\). However, not all quadratics include a middle term, like our special case, \(100x^2 - 121\), which is a pure difference of squares.
Quadratics can represent areas, motion, economics, and much more, thus making understanding them essential to various fields.
  • Recognizing expressions as quadratics helps in employing the right method to solve or factor them effectively.

  • Quadratics can be solved using factoring, completing the square, or the quadratic formula, making flexibility in approach important.

Getting comfortable with quadratics and their properties helps improve problem-solving speed, and it builds a strong foundation in algebra.

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Most popular questions from this chapter

Find the greatest common factor of the numbers. 9 and 12

Factor the expression. Tell which special product factoring pattern you used. $$169-x^{2}$$

Use factoring to solve the equation. Use a graphing calculator to check your solution if you wish. $$\frac{1}{3} x^{2}-6 x+27=0$$

Simplify the expression. $$\sqrt{\frac{28}{49}}$$

Sketch the graph of the inequality. \(y-3 x \geq 2\)
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