Problem 11 - Intermediate Algebra

Question

Solve each of the quadratic equations by factoring and applying the property, \(a b=0\) if and only if \(a=0\) or \(b=0\). If necessary, return to Chapter 3 and review the factoring techniques presented there. $$ x^{2}-19 x+84=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = 12\) and \(x = 7\).

Step 1: Identify the equation

The quadratic equation given is \(x^2 - 19x + 84 = 0\). We need to solve it by factoring and using the zero-product property.

Step 2: Factor the quadratic equation

To factor \(x^2 - 19x + 84 = 0\), we need two numbers that multiply to 84 and add up to 19. These numbers are 12 and 7, so we can write the equation as \((x - 12)(x - 7) = 0\).

Step 3: Apply the zero-product property

According to the zero-product property, if \(a \cdot b = 0\), then either \(a = 0\) or \(b = 0\). So, we set each factor equal to 0: 1. \(x - 12 = 0\) 2. \(x - 7 = 0\)

Step 4: Solve for \(x\)

Solve each equation from Step 3. 1. For \(x - 12 = 0\), add 12 to both sides to get \(x = 12\). 2. For \(x - 7 = 0\), add 7 to both sides to get \(x = 7\).

Key Concepts

Factoring Techniques

Factoring is a key technique when solving quadratic equations. It's the process of breaking down an equation into simpler expressions (factors) that, when multiplied together, give the original equation. To factor a quadratic equation such as \(x^2 - 19x + 84 = 0\), you need to find two numbers that multiply to the constant term (84) and add up to the linear coefficient (-19).

A useful tip is to list the factor pairs of the constant term and test which pair adds up to the linear coefficient. In this case:

• The factor pairs of 84 are (1, 84), (2, 42), (3, 28), (4, 21), (6, 14), and (7, 12).
• Among these, the numbers 12 and 7 multiply to 84 and add to 19.

Using these numbers, we rewrite the quadratic expression as \((x - 12)(x - 7) = 0\).

This factoring down simplifies the equation, making it easier to solve.

Zero-Product Property

The zero-product property is an important concept in algebra. It states that if a product of two numbers is zero, then at least one of the multiplicands must be zero. Mathematically, if \(a \cdot b = 0\), then \(a = 0\) or \(b = 0\).

When solving the factored form of a quadratic equation like \((x - 12)(x - 7) = 0\), apply this property to set each factor equal to zero.

• Set \(x - 12 = 0\)
• Set \(x - 7 = 0\)

Once you identify that one of the factors must be zero, you can proceed to solving simple equations step-by-step. This property effectively breaks down a complex equation into manageable parts, simplifying the process of finding solutions.

Solving Quadratic Equations

Solving quadratic equations involves finding the values of \(x\) that satisfy the equation. Once you have factored the quadratic to get \((x - 12)(x - 7) = 0\), the next steps become straightforward because of the zero-product property.

To solve the equation, take the two linear equations derived from setting each factor to zero:

• For \(x - 12 = 0\), add 12 to both sides to get \(x = 12\).
• For \(x - 7 = 0\), add 7 to both sides to get \(x = 7\).

Thus, the solutions to the quadratic equation \(x^2 - 19x + 84 = 0\) are \(x = 12\) and \(x = 7\).

By correctly applying factoring and the zero-product property, you can efficiently solve any quadratic equation. Practice factoring different quadratic equations to become more comfortable with this process.