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Chapter 1: Problem 14

Solve each equation by factoring. $$ x^{2}-9=0 $$

Short Answer

Expert verified
x = 3 and x = -3

Step by step solution

01

- Recognize the Form of the Equation

Notice that the equation is in the form of a difference of squares: \[ x^{2} - 9 = 0 \] This can be simplified because it can be expressed as \[ a^{2} - b^{2} = (a - b)(a + b) \]
02

- Factor the Difference of Squares

Rewrite the equation as the product of two binomials: \[ x^{2} - 9 = (x - 3)(x + 3) \]
03

- Set Each Factor Equal to Zero

Set each factor to zero and solve for x:\[x - 3 = 0\] and \[x + 3 = 0\]
04

- Solve the Equations

Solve the individual equations: \( x - 3 = 0 \) gives \( x = 3 \)\( x + 3 = 0 \) gives \( x = -3 \)

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Key Concepts

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

difference of squares
In algebra, the 'difference of squares' refers to a specific type of binomial that takes the form \(a^2 - b^2\). This form is unique because it can be factored into the product of two simpler binomials.
Notice that \[(a^2 - b^2) = (a - b)(a + b)\]
Examples include:
\(x^2 - 9 = (x - 3)(x + 3)\)
\(y^2 - 16 = (y - 4)(y + 4)\)
Recognizing this form helps simplify solving quadratic equations.
solving quadratic equations
A quadratic equation is any equation that can be written in the form \(ax^2 + bx + c = 0\). In this form, \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable.
There are various methods to solve quadratic equations:
  • Factoring
  • Using the quadratic formula
  • Completing the square
  • Graphing

For our specific problem, since it is in a 'difference of squares' form, we use factoring.
factoring binomials
Factoring binomials involves writing a binomial expression as a product of simpler expressions. For the difference of squares:
\(a^2 - b^2 = (a - b)(a + b)\)
It's important to identify when an expression fits this form. In our problem:
\(x^2 - 9 = (x - 3)(x + 3)\)
By recognizing the form, we simplify the equation.
algebraic equations
Algebraic equations consist of numbers, symbols, and variables connected by operators. Solving algebraic equations involves finding the value of the variables that make the equation true.
There are multiple types of algebraic equations:
  • Linear equations
  • Quadratic equations
  • Polynomial equations

In this exercise, we focused on a quadratic equation using the difference of squares method to solve it.

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