Problem 11 - College Algebra

Question

Solve the quadratic equation by factoring. $$ 3 x^{2}-75=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = -5\) and \(x = 5\).

Step 1: Set the equation equal to zero

The equation is already given as \(3x^2 - 75 = 0\), so there is no need to make any changes here. We can move on to the next step.

Step 2: Factor out the Greatest Common Factor

Identify the greatest common factor (GCF) of the terms \(3x^2\) and \(-75\). The GCF is 3. Factor out the 3 from the equation:\[ 3(x^2 - 25) = 0 \]

Step 3: Recognize the difference of squares

The expression inside the parentheses, \(x^2 - 25\), is a difference of squares. It can be factored as:\[ (x + 5)(x - 5) \]

Step 4: Write the fully factored equation

Now substitute back the factored expression of the difference of squares into the equation:\[ 3(x + 5)(x - 5) = 0 \]

Step 5: Apply the Zero Product Property

According to the zero product property, if a product of factors equals zero, then at least one of the factors must be zero. Set each factor equal to zero:1. \(x + 5 = 0\)2. \(x - 5 = 0\)

Step 6: Solve each factor equation

Solve the two simple equations obtained from the previous step:1. \(x + 5 = 0\) gives \(x = -5\).2. \(x - 5 = 0\) gives \(x = 5\).

Step 7: Write the solution set

The solutions to the quadratic equation are \(x = -5\) and \(x = 5\). These are the values of \(x\) that satisfy the original equation.

Key Concepts

Factoring

Factoring is a method used to express a polynomial as a product of its factors. It simplifies expressions and makes it easier to solve equations. In the context of a quadratic equation, like the one given, factoring involves expressing the quadratic expression as a product of binomials.

To factor a quadratic equation:

• First identify the greatest common factor (GCF) if there is any. This step simplifies the equation by reducing it to its simplest form.
• Next, rewrite the quadratic expression as a product of binomial expressions. This requires understanding the structure of the quadratic expression—determining if it fits patterns like trinomial or difference of squares.

By factoring the equation, we simplify the process of finding the roots, making the entire process more straightforward than using other methods such as completing the square or using the quadratic formula.

Difference of Squares

The difference of squares is a specific type of expression that takes the form: \[a^2 - b^2\]Such expressions can be factored with a special rule: \[(a + b)(a - b)\]

This method can be applied when the quadratic expression can be rewritten as the subtraction of two perfect squares. In the example \[x^2 - 25\], both terms are perfect squares, so \[x^2 - 25\] becomes \[(x + 5)(x - 5)\].

• The term \(x^2\) is the square of \(x\).
• The number 25 is the square of 5.

By recognizing expressions as a difference of squares, we can quickly and accurately factor them, which is especially useful in solving quadratic equations by factoring. This technique simplifies the algebraic process significantly.

Zero Product Property

The Zero Product Property is a fundamental principle in algebra. It states that if a product of two or more factors is zero, then at least one of the factors must be zero. This property is used to find the solutions of a factored equation

In the factored equation:\[3(x + 5)(x - 5) = 0\]we apply the zero product property by setting each factor equal to zero:

• \(x + 5 = 0\)
• \(x - 5 = 0\)

Solving each equation individually gives the potential solutions of the quadratic equation. This property is crucial for finding solutions efficiently once an expression is fully factored, as it breaks down the solution process into simpler linear equations.

Solutions of Quadratic Equations

Finding the solutions of a quadratic equation means identifying the values of \(x\) that satisfy the equation. These values are also known as the "roots" of the equation.

Once a quadratic equation is factored, the solutions can be found by applying the zero product property. For the example given, \[3(x + 5)(x - 5) = 0\], we set each factor to zero:

• \(x + 5 = 0\) results in \(x = -5\)
• \(x - 5 = 0\) results in \(x = 5\)

These solutions indicate the points where the quadratic graph intersects the x-axis. They are the roots of the polynomial. By understanding these roots, you gain insight into the behavior of the quadratic equation, such as vertex location and axis of symmetry.