Problem 79 - Elementary Algebra

Question

For the following problems, solve the equations, if possible. $$ 18 b^{2}+24 b+6=0 $$

Step-by-Step Solution

Answer

Answer: The roots of the given quadratic equation are \(b = \frac{-1}{3}\) and \(b = -1\).

Step 1: 1. Factor out the greatest common divisor (GCD)

First, observe the coefficients of the quadratic equation: 18, 24, and 6. Find the GCD of these coefficients. The GCD of 18, 24, and 6 is 6. Factor out 6 from the equation: $$ 6(3b^2 + 4b + 1) = 0 $$

Step 2: 2. Factor the quadratic expression

Now, factor the quadratic expression inside the parentheses as a product of two binomials. Observe that the constants of the binomials must multiply to 1 and their linear terms must add up to 4b. The factoring is as follows: $$ (3b + 1)(b + 1) = 0 $$

Step 3: 3. Apply the Zero-Product Property

When a product of factors is equal to zero, it means that at least one of the factors must be equal to zero. So, we can write two separate equations, setting each factor equal to zero: $$ 3b + 1 = 0 \text{ or } b + 1 = 0 $$

Step 4: 4. Solve for b

Solve each equation for b: $$ \text{For } 3b + 1 = 0, \\ 3b = -1 \\ b = \frac{-1}{3} $$ and $$ \text{For } b + 1 = 0, \\ b = -1 $$ The roots of the quadratic equation are b = \(\frac{-1}{3}\) and b = -1.

Key Concepts

Factoring

Factoring is a mathematical process used to break down expressions into simpler components. In the context of quadratic equations like \(18b^2 + 24b + 6 = 0\), factoring involves rewriting the quadratic expression into a product of two simpler expressions, usually binomials, that when multiplied give the original.
To start factoring a quadratic equation, we first look for patterns or common terms. Factoring can simplify the equation significantly, making it easier to solve. For instance, in the expression \(3b^2 + 4b + 1\), we need to find numbers that multiply to the constant term (1) and add up to the linear coefficient (4).

• For \(3b^2 + 4b + 1\), observe that:\((3b + 1)(b + 1) = 3b^2 + 4b + 1\).
• Each term in the binomials multiplies to give us the original quadratic expression.

In summary, factoring helps to rewrite quadratics into their simpler forms and is a crucial step in solving quadratic equations efficiently.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that divides all the coefficients of an equation without leaving a remainder. In the polynomial equation \(18b^2 + 24b + 6 = 0\), identifying the GCD helps in simplifying the equation by scaling down the coefficients.
Finding the GCD involves:

• Listing the factors of each coefficient: 18, 24, and 6.
• The factors of 18 are 1, 2, 3, 6, 9, 18.
• The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24.
• The factors of 6 are 1, 2, 3, 6.
• The GCD is the largest common factor from all lists, which is 6.

Factoring out the GCD simplifies the calculation and makes it easier to handle the quadratic equation. Thus, the equation becomes \(6(3b^2 + 4b + 1) = 0\), where the complexities of larger numbers are reduced.

Zero-Product Property

The Zero-Product Property is an algebraic property that states if a product of two or more factors is zero, then at least one of the factors must be zero. This property is essential when solving factored quadratic equations.
Consider the factored equation \((3b + 1)(b + 1) = 0\). According to the zero-product property, for the product to be zero, either:

• \(3b + 1 = 0\)
• or \(b + 1 = 0\)

Using this property allows us to break down complex equations into simple linear equations that are much easier to solve. By setting each factor to zero, we systematically find the solutions of the quadratic equation.

Solving Equations

Solving equations involves finding the values of the variables that make the equation true. Once a quadratic equation is factored and separated using the zero-product property, solving these simple linear equations becomes straightforward.
For the factors \(3b + 1\) and \(b + 1\), set each equal to zero:

• From \(3b + 1 = 0\), we solve for \(b\):
  Subtract 1 from both sides gives \(3b = -1\).
  Divide both sides by 3 gives \(b = \frac{-1}{3}\).
• From \(b + 1 = 0\), we solve for \(b\):
  Subtract 1 from both sides gives \(b = -1\).

The solutions \(b = \frac{-1}{3}\) and \(b = -1\) satisfy the original equation, demonstrating the effectiveness of systematically applying these methods for solving quadratic equations.