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Chapter 10: Problem 1

What does it mean to factor a quadratic expression?

Short Answer

Expert verified
Factoring a quadratic expression means breaking it down into simpler binomial expressions that, when multiplied together, result in the original quadratic expression. For example, \(x^2 + 5x + 6\) can be factored into \((x+2)(x+3)\).

Step by step solution

01

Understand the structure of a quadratic expression

A quadratic expression typically takes the form \(ax^2 + bx + c\), where a, b, and c are constants. Recognizing this structure is the first step toward being able to factor the expression. However, it should be noted that the coefficients do not need to be numerical, but can be any constant expression.
02

Identify the goal

Factoring a quadratic expression means breaking it down into simpler binomial expressions that, when multiplied together, result in the original quadratic expression. This is, in essence, decomposition.
03

Factoring process

Start by looking for two numbers that multiply to give ac (the product of a and c) and add to give b. These numbers replace the middle term. Then, factor by grouping. Lastly, rewrite the expression as the product of two binomial expressions. For example, the quadratic expression \(x^2 + 5x + 6\) can be factored into \((x+2)(x+3)\).

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Key Concepts

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

Quadratic expression
A quadratic expression is a mathematical expression that includes a variable raised to the second power. The general form is \(ax^2 + bx + c\) where:
  • \(a\), \(b\), and \(c\) are constants.
  • \(x\) represents the variable.
  • \(a eq 0\) since without \(x^2\), it's not a quadratic expression.
Understanding this form helps in identifying a quadratic and preparing it for factoring. Quadratics play a crucial role in algebra due to their appearance in various equations and functions.
Quadratic expressions show up in multiple scenarios, such as:
1. Solving quadratic equations.
2. Graphing parabolas.
3. Modeling real-world problems like projectile motion.
Recognizing a quadratic expression is the first step in manipulating it for solutions or modeling.
Binomial expressions
Binomial expressions are algebraic expressions that contain just two terms, separated by a plus or minus sign. Binomials are often written in the form \((x + p)(x + q)\) where:
  • Each set of parentheses contains two terms.
  • \(p\) and \(q\) are constants.
Binomials are essential when factoring quadratic expressions. When a quadratic is successfully factored, it is expressed as the product of two binomial expressions.
This means instead of one complex quadratic, you have two simpler expressions that you can work with individually. For example, the quadratic \(x^2 + 5x + 6\) can be expressed as the product of two binomials: \((x + 2)(x + 3)\).
Binomial expressions are easier to handle and simplify many problems in algebra.
Factoring by grouping
Factoring by grouping is a method used when a quadratic expression is not easily factorable by simple inspection. This technique breaks down the process into manageable steps:
  • First, find two numbers that multiply to give \(ac\) (the product of the first and last coefficients) and add to give \(b\) (the middle coefficient).
  • Rewrite the middle term using these two numbers.
  • "Group" the terms in pairs and factor out the common factor in each group.
  • Finally, express the expression as a product of two binomials.
For instance, with \(x^2 + 5x + 6\), find numbers that multiply to 6 (from \(1 \times 6\)) and add to 5; these are 2 and 3. Rewrite as \(x^2 + 2x + 3x + 6\), then group: \(x(x+2) + 3(x+2)\), giving \((x+3)(x+2)\).
Factoring by grouping helps in transforming a seemingly complex quadratic into recognizable binomials quickly, making it a powerful tool in algebra.

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Most popular questions from this chapter

Factor the expression. Tell which special product factoring pattern you used. $$-2 x^{2}+52 x-338$$

Find the product. $$(b+9)(b-9)$$

Which one of the following equations cannot be solved by factoring with integer coefficients? (A) \(12 x^{2}-15 x-63=0\) (B) \(12 x^{2}+46 x-8=0\) (C) \(6 x^{2}-38 x-28=0\) (D) \(8 x^{2}-49 x-68=0\)

Simplify the expression. $$\frac{12 \sqrt{4}}{\sqrt{9}}$$

Which one of the following is a correct factorization of the expression \(-16 x^{2}+36 x+52 ?\). A) \((-16 x+26)(x-2)\) (B) \((-4 x-13)(4 x+4)\) (C) \(-1(4 x-13)(4 x+4)\) (D) \(-1(4 x-4)(4 x+13)\)
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