Problem 15 - Elementary and Intermediate Algebra

Question

a. Suppose we wish to factor \(12 b^{2}+20 b-9\) by grouping. Identify \(a, b,\) and \(c\) b. What is the key number, \(a c ?\)

Step-by-Step Solution

Answer

a\) a = 12, b = 20, c = -9; b\) ac = -108.

Step 1: Identify the Quadratic Equation Form

The given expression is \(12b^2 + 20b - 9\). We need to identify the coefficients in the standard form of a quadratic equation, which is \(ax^2 + bx + c\).

Step 2: Identify Coefficients a, b, and c

Here, the coefficient of \(b^2\) is \(a=12\), the coefficient of \(b\) is \(b=20\), and the constant term is \(c=-9\).

Step 3: Calculate the Key Number ac

To find the key number, multiply \(a\) and \(c\):\[ ac = 12 \times (-9) = -108 \]

Key Concepts

Quadratic Equation Coefficients

Quadratic equations are polynomials of the second degree, typically expressed in the standard form: \(ax^2 + bx + c\). Here, each of the terms has a specific role. The value \(a\) is known as the coefficient of the quadratic term \(x^2\), \(b\) is the coefficient of the linear term \(x\), and \(c\) is the constant term. Recognizing these coefficients is fundamental to solving quadratic equations through methods like factoring.

For the equation presented, \(12b^2 + 20b - 9\), you can identify:

• \(a = 12\)
• \(b = 20\)
• \(c = -9\)

Understanding these coefficients is essential because they determine the equation's behavior and how you tackle the factoring process.

Key Number in Factoring

The concept of the key number arises primarily in the factoring of quadratic equations through grouping. It involves the product of the quadratic coefficient \(a\) and the constant term \(c\). This product helps identify two numbers that can be used to decompose the middle term, \(b\), to apply the grouping method effectively.

Essentially, you calculate the key number using: \[ ac = a \times c \] For our equation, \(ac\) becomes: \[ 12 \times (-9) = -108 \]

• This key number helps transform the quadratic equation into a form that might be easier to factor by identifying pairs of numbers whose sum equals the middle coefficient \(b\) and whose product equals \(ac\).

Grouping Method for Factoring

Once the key number has been identified, the next strategy is using the grouping method to factor the quadratic equation. This technique involves splitting the middle term into two terms whose coefficients multiply to the key number and add up to the original middle term coefficient \(b\). This approach allows us to eventually group terms to make factoring possible.

Here is the process in a nutshell:

• Split the middle term using two numbers that multiply to \(ac\) and add up to \(b\).
• Rewrite the equation with these two terms.
• Group the terms into pairs and factor out the common factors in each group.
• Combine the groups to achieve a fully factored form of the equation.

Each of these steps helps to systematically simplify and eventually solve the equation by identifying common factors and utilizing the distributive property for algebraic expressions.