Problem 34 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ a^{2}-17 a+60=0 $$

Step-by-Step Solution

Answer

The solutions are \( a = 12 \) and \( a = 5 \).

Step 1: Identify the type of equation and components

The given equation is a quadratic equation in the form of \( ax^2 + bx + c = 0 \). Here, it's \( a^2 - 17a + 60 = 0 \) with \( a = 1 \), \( b = -17 \), and \( c = 60 \). We need to solve for the variable \( a \).

Step 2: Confirm if factorizable by product method

To solve the equation by factoring, we look for two numbers that multiply to \( c \) (which is 60) and add up to \( b \) (which is -17).

Step 3: Identify the correct factor pair

The numbers \(-12\) and \(-5\) multiply to 60 and add to -17, making them the correct pair to factor the quadratic as \((a - 12)(a - 5) = 0\).

Step 4: Solve each factor for \(a\)

Set each factor equal to zero: 1. \( a - 12 = 0 \) implies \( a = 12 \).2. \( a - 5 = 0 \) implies \( a = 5 \).

Step 5: Conclude the solutions

The solutions are the values of \( a \) that satisfy the original equation. Thus, the solutions are \( a = 12 \) and \( a = 5 \).

Key Concepts

Factoring

Factoring is a key technique used to solve quadratic equations by expressing them as a product of two binomial expressions. This method is particularly useful when the quadratic equation can be decomposed into two factors that are easily identifiable. In the exercise, we identified the quadratic equation:
\[ a^2 - 17a + 60 = 0 \]This form is known as standard form, where it’s expressed as \( ax^2 + bx + c = 0 \). Factoring involves finding two numbers that multiply to the constant term \( c \) (in this case, 60) and add up to the linear coefficient \( b \) (which is -17). The numbers

• -12 and -5

were chosen because:

• -12 × -5 = 60
• -12 + -5 = -17

This allows us to rewrite the equation as:\[ (a - 12)(a - 5) = 0 \]Setting each factor equal to zero will later enable us to find the solutions for the variable.

Quadratic Formula

Although the exercise focuses on factoring, the quadratic formula is another powerful tool to solve quadratic equations, especially when factoring is not straightforward. The quadratic formula is:\[ a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Where:

• \(a\) is the coefficient of \(x^2\)
• \(b\) is the coefficient of \(x\)
• \(c\) is the constant term

This formula provides an alternative to factoring, offering a direct path to find solutions for any quadratic equation. In the context of our original equation, it can be used if factoring is not feasible or when the roots are not rational numbers.
What's powerful about the quadratic formula is that it utilizes the discriminant \(b^2 - 4ac\), which helps determine the nature of the roots (real or complex).

Solutions of Equations

Finding the solutions of an equation such as a quadratic means identifying the values that satisfy the equation, essentially making it true. In this exercise, solving \[ (a - 12)(a - 5) = 0 \] involved applying the zero product property. This important concept states that if the product of two terms is zero, then at least one of the terms must be zero:

• If \(a - 12 = 0\), then \(a = 12\).
• If \(a - 5 = 0\), then \(a = 5\).

Thus, the solutions to the equation are \(a = 12\) and \(a = 5\). These solutions mean substituting either 12 or 5 back into the original equation results in a valid equality, confirming their correctness.
Determining the solutions of quadratic equations can be done either by factoring, as detailed, or using other methods like completing the square or the quadratic formula for confirmation or more complex scenarios.