Problem 13 - College Algebra with Corequisite Support

Question

For the following exercises, solve the quadratic equation by factoring. $$ 4 x^{2}=9 $$

Step-by-Step Solution

Answer

The solutions are \( x = \frac{3}{2} \) and \( x = -\frac{3}{2} \).

Step 1: Move all terms to one side of the equation

To solve the equation by factoring, we first need to set the equation to zero. Subtract 9 from both sides of the equation: \[ 4x^2 - 9 = 0 \]

Step 2: Recognize the difference of squares

The equation \( 4x^2 - 9 = 0 \) is a difference of squares. It can be written as: \[ (2x)^2 - 3^2 = 0 \]

Step 3: Apply the difference of squares formula

Using the difference of squares formula \( a^2 - b^2 = (a-b)(a+b) \), factor the equation: \[ (2x - 3)(2x + 3) = 0 \]

Step 4: Solve each factor set to zero

Set each factor equal to zero and solve for \( x \): For \( 2x - 3 = 0 \):\[ 2x = 3 \]\[ x = \frac{3}{2} \]For \( 2x + 3 = 0 \):\[ 2x = -3 \]\[ x = -\frac{3}{2} \]

Step 5: State the solutions

The solutions for the quadratic equation \( 4x^2 = 9 \) are: \[ x = \frac{3}{2} \] and \[ x = -\frac{3}{2} \]

Key Concepts

Factoring

Factoring is a fundamental skill in algebra that involves expressing a mathematical expression as a product of its factors. When it comes to quadratic equations, factoring is a powerful method used for finding the solutions or "roots" of the equation. The process requires rearranging a quadratic equation into a factorable form, usually achieved by making one side of the equation equal to zero. Once in this form, terms can be grouped and decomposed into their simplest product terms.

For example:

• Consider the equation: \( x^2 - 5x + 6 = 0 \).
• Here, factoring the quadratic gives: \((x - 2)(x - 3) = 0\).
• This indicates that \( x = 2 \) and \( x = 3 \) are solutions of the equation.

Breaking down expressions into these simpler components not only reveals the solutions but also offers insight into the behavior of the quadratic function.

Difference of squares

The difference of squares is a special factoring technique used specifically with expressions of the form \( a^2 - b^2 \). This form can be factored into \( (a-b)(a+b) \). It's a handy tool for solving quadratic equations where you spot terms that fit this specific pattern.

In the equation \( 4x^2 - 9 = 0 \):

• This can be rewritten as \( (2x)^2 - 3^2 = 0 \).
• This clearly shows it's the difference of squares.
• Apply the formula to factor it as \( (2x - 3)(2x + 3) = 0 \).

By applying this technique, we simplify the quadratic to a product of two binomial terms, making it easier to identify and solve the solutions.

Equation solving

Equation solving is the process of finding the values of the variable that make the equation true. In the context of quadratic equations, this typically involves isolating the variable through various algebraic techniques, including factoring, completing the square, or using the quadratic formula.

For our specific equation after factoring, \( (2x - 3)(2x + 3) = 0 \), the next step is solving each part:

• Set \( 2x - 3 = 0 \) and solve for \( x \). This gives \( x = \frac{3}{2} \).
• Set \( 2x + 3 = 0 \) and solve for \( x \). This gives \( x = -\frac{3}{2} \).

The solutions to the equation are where these two binomial parts equal zero. By solving the simpler equations derived from factoring, one finds the roots of the original quadratic equation.

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. It is the underlying framework for solving equations, expressing unknown values, and understanding patterns and relationships in numbers.

Quadratic equations, a staple in algebra, are polynomial equations of degree two and have the general form \( ax^2 + bx + c = 0 \). The power and universality of algebra allow us to represent complex mathematical relationships and solve them systematically. By applying algebraic concepts such as factoring and recognizing patterns like the difference of squares, we can tackle problems efficiently and find meaningful solutions.

Embracing algebraic techniques gives us a toolkit for addressing various types of problems, not only quadratic equations but a wide array of mathematical expressions.