Problem 52 - Precalculus Mathematics for Calculus

Question

Solve the equation by factoring. $$3 x^{2}-27=0$$

Step-by-Step Solution

Answer

The solutions are \( x = -3 \) and \( x = 3 \).

Step 1: Set the Equation to Equal Zero

Start by ensuring the equation is set to zero, which in this case, it already is: \[ 3x^2 - 27 = 0 \]

Step 2: Factor Out the Greatest Common Factor (GCF)

Identify the greatest common factor in the equation. Here, the GCF is 3. Factor out 3 from each term:\[ 3(x^2 - 9) = 0 \]

Step 3: Apply the Difference of Squares

Notice that \( x^2 - 9 \) is a difference of squares. It can be rewritten as:\[ (x + 3)(x - 3) = 0 \]

Step 4: Solve Each Factor for Zero

Set each factor equal to zero and solve for \( x \):\( x + 3 = 0 \) gives \( x = -3 \)\( x - 3 = 0 \) gives \( x = 3 \)

Step 5: State the Solutions

The solutions to the equation are the values of \( x \) that satisfy each individual equation from Step 4. Thus, the solutions are:\[ x = -3 \] and \[ x = 3 \]

Key Concepts

Difference of Squares

The "difference of squares" is a special factoring technique used in quadratic equations. It centers around expressions where two squared terms are subtracted, such as \( x^2 - 9 \). The format for a difference of squares is always \( a^2 - b^2 \), where both \( a \) and \( b \) are squared terms. This form can be factored into two binomials:

• \( (a + b)(a - b) \)

For our expression \( x^2 - 9 \), we recognize that it is a difference of squares, since \( 9 \) can be rewritten as \( 3^2 \). Thus, \( x^2 - 9 \) can be factored as:

• \( (x + 3)(x - 3) \)

This method is an efficient way to simplify and solve equations by breaking them into simpler components, which can then be easily solved individually.

Greatest Common Factor

The "greatest common factor" (GCF) is a key concept in algebra which aids in simplifying equations before tackling more complex factoring techniques. The GCF is the largest number that divides each term of the expression without leaving a remainder. When attempting to factor an equation like \( 3x^2 - 27 = 0 \), the GCF is essential for making the expression easier to work with. In this example, the coefficients of the terms are 3 and 27.

• Finding the GCF involves identifying the largest number that can evenly divide both 3 and 27.

That number is 3, as it divides both terms without a remainder. We factor it out to simplify the equation from:

• \( 3(x^2 - 9) = 0 \)

By factoring out the GCF, we simplify the quadratic and can subsequently apply other factoring techniques like the difference of squares.

Solving Equations

Once an equation is factored, solving it becomes a matter of solving each factor individually. After factoring \( 3(x^2 - 9) \) into \( 3(x + 3)(x - 3) = 0 \), the next step is setting each factor that contains a variable equal to zero:

• \( x + 3 = 0 \)
• \( x - 3 = 0 \)

Solving these gives the solutions for \( x \). For \( x + 3 = 0 \), subtract 3 from both sides and you find that:

• \( x = -3 \)

Similarly, solve \( x - 3 = 0 \) to find:

• \( x = 3 \)

The solutions, \( x = -3 \) and \( x = 3 \), indicate the points where the original quadratic equation equals zero. This method is typical for solving quadratic equations post-factoring.

Mathematical Factoring Techniques

Mathematical factoring techniques are strategies to break down polynomials into products of simpler expressions. They are vital in simplifying expressions and solving equations. The process relies on recognizing patterns like the greatest common factor and difference of squares, which we used in solving \( 3x^2 - 27 = 0 \). Key techniques include:

• Removing the GCF: Start by removing the greatest common factors to simplify the polynomial.
• Difference of Squares: Recognize patterns where squares are subtracted, enabling them to be factored into two binomials.

These techniques transform complex equations into simpler, solvable forms. Understanding each method and when to use it aids in efficiently solving polynomial equations and mastering algebra.