Problem 44 - Brief Applied Calculus

Question

Solve each equation by factoring or the Quadratic Formula, as appropriate. $$ 3 x^{2}-27=0 $$

Step-by-Step Solution

Answer

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

Step 1: Write the equation in standard form

Start by writing the given equation in its standard quadratic form, which is \( ax^2 + bx + c = 0 \). The given equation is \( 3x^2 - 27 = 0 \). Since it is already in the standard form of \( ax^2 + bx + c = 0 \) with \( a = 3 \), \( b = 0 \), and \( c = -27 \), we can proceed to the next step.

Step 2: Factor the quadratic equation

To factor the equation \( 3x^2 - 27 = 0 \), first factor out the greatest common factor from the terms. Both terms, \( 3x^2 \) and \( -27 \), can be divided by 3, so we have:\[ 3(x^2 - 9) = 0 \]Next, factor \( x^2 - 9 \) using the difference of squares formula, \( a^2 - b^2 = (a+b)(a-b) \). Here, \( x^2 - 9 \) becomes \((x + 3)(x - 3)\). The equation now is:\[ 3(x + 3)(x - 3) = 0 \]

Step 3: Apply the Zero Product Property

Use the Zero Product Property, which states that if the product of factors is zero, then at least one of the factors must be zero. Set each factor to zero:1. \( x + 3 = 0 \) 2. \( x - 3 = 0 \) Solve each equation for \( x \).

Step 4: Solve each equation

Solve the equations from Step 3:1. \( x + 3 = 0 \) gives \( x = -3 \). 2. \( x - 3 = 0 \) gives \( x = 3 \). These are the solutions to the original equation.

Key Concepts

Factoring

Factoring is a simple and effective method used to solve quadratic equations by expressing them as a product of simpler expressions. To factor a quadratic equation like \( 3x^2 - 27 = 0 \), first look for a common factor in each term. Here, both \( 3x^2 \) and \(-27\) can be divided by 3, giving us:

• Factor out the greatest common factor: \( 3(x^2 - 9) = 0 \).
• Recognize the expression inside the parentheses, \( x^2 - 9 \), as a difference of squares.
• Applying the difference of squares formula, we get \((x + 3)(x - 3)\).

The factored equation is \( 3(x + 3)(x - 3) = 0 \). The zero product property tells us that if the product of factors equals zero, then at least one factor must be zero. Solving \( x + 3 = 0 \) and \( x - 3 = 0 \) gives us the solutions \( x = -3 \) and \( x = 3 \). Factoring can be a quick and neat method when the quadratic term is easily factorable like in this case.

Quadratic Formula

The quadratic formula is a universal method for solving any quadratic equation, regardless of whether it can be easily factored or not. It is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Where \( a \), \( b \), and \( c \) are coefficients from the quadratic equation in standard form \( ax^2 + bx + c = 0 \).In our original problem, \( 3x^2 - 27 = 0 \), we identify \( a = 3 \), \( b = 0 \), and \( c = -27 \). Applying these values into the quadratic formula:

• Calculate the discriminant: \( b^2 - 4ac = 0^2 - 4(3)(-27) = 324 \).
• The formula becomes: \( x = \frac{0 \pm \sqrt{324}}{6} \).
• Simplifying gives us \( x = \frac{\pm 18}{6} \).

This yields the solutions \( x = 3 \) and \( x = -3 \). The quadratic formula is particularly useful when the quadratic equation is not easily factorable, providing a straightforward approach to finding roots.

Difference of Squares

The difference of squares is a specific factoring pattern that appears frequently in solving quadratic equations. It's written as:\[ a^2 - b^2 = (a + b)(a - b) \]This formula is extremely handy because it allows us to rewrite expressions in a multiplicative form which is often easier to solve.In our problem \( 3x^2 - 27 = 0 \), after factoring out 3, we observe \( x^2 - 9 \), which can be factored using this pattern:

• Recognize \( x^2 \) as \( a^2 \) where \( a = x \).
• Recognize \( 9 \) as \( b^2 \) where \( b = 3 \).
• Apply the difference of squares: \( x^2 - 9 = (x + 3)(x - 3) \).

Using the difference of squares simplifies the equation into a product of two binomials, \( (x + 3)(x - 3) \), allowing for easy application of the zero product property to solve for \( x \). Understanding the difference of squares is an important algebraic technique for quickly finding solutions in quadratic equations that suit this pattern.