Problem 96 - Algebra for College Students

Question

Explain your thought process when factoring \(30 x^{2}+13 x-56\)

Step-by-Step Solution

Answer

The factored form of \(30x^2 + 13x - 56\) is \((6x - 7)(5x + 8)\).

Step 1: Identify Coefficients

We start by identifying the coefficients of the quadratic equation. The equation is \(30x^2 + 13x - 56\). The coefficient \(a\) for \(x^2\) is 30, the coefficient \(b\) for \(x\) is 13, and the constant term \(c\) is -56.

Step 2: Find Two Numbers Based on AC Method

Multiply \(a\) and \(c\): \(30 \times (-56) = -1680\). We need to find two numbers that multiply to -1680 and add to 13, the coefficient \(b\). These numbers are 48 and -35, because \(48 \times (-35) = -1680\) and \(48 + (-35) = 13\).

Step 3: Rewrite Middle Term

Using the numbers from Step 2, rewrite the middle term of the equation: \[30x^2 + 48x - 35x - 56\]. This splits the equation into four terms.

Step 4: Factor by Grouping

Group the terms in pairs and factor each pair: \[(30x^2 + 48x) + (-35x - 56)\]. Factor out the common factors: \[6x(5x + 8) - 7(5x + 8)\]. Notice the common binomial \((5x + 8)\).

Step 5: Factor Out Common Binomial

Factor out the common binomial \((5x + 8)\) from the expression: \( (6x - 7)(5x + 8) \). This gives us the factored form of the original quadratic expression.

Key Concepts

AC method

The AC Method is a technique used for factoring quadratic equations when the leading coefficient is not 1. In our example, we have the quadratic equation \(30x^2 + 13x - 56\). The AC method is so named because you multiply \(a\) and \(c\), the coefficients of the first and last terms. The product you get is the basis for finding two numbers that, when multiplied, give you the product \(ac\) and, when added, yield the middle coefficient \(b\).Let's break down these steps:

• Identify \(a = 30\) and \(c = -56\).
• Calculate \(a \times c = 30 \times (-56) = -1680\).
• Find two numbers that multiply to -1680 and add to 13. These numbers are 48 and -35.

By using the AC Method, we can systematically break down the quadratic equation into simpler terms that can be factored more easily in the next steps.

factoring by grouping

Factoring by grouping is a method that comes into play after you've rewritten a quadratic expression with four terms. It's a process of regrouping terms to factor them into pairs.To start, rewrite the quadratic equation. Following our example:\[30x^2 + 48x - 35x - 56\].Here, the middle term \(13x\) has been split into \(48x\) and \(-35x\) based on the numbers we found using the AC method.Now group these terms:

• Group: \((30x^2 + 48x) + (-35x - 56)\).
• Factor each group: Extract common factors, which gives us \(6x(5x + 8) - 7(5x + 8)\).

Notice that the expression now involves a common binomial, \((5x + 8)\). Factoring by grouping transforms the original quadratic into a product of binomials, making it much easier to manage and solve.

quadratic equation

A quadratic equation is a second-degree polynomial equation in a single variable \(x\). The general form is \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). This equation can be solved using various methods such as factoring, completing the square, or applying the quadratic formula.For the given quadratic equation \(30x^2 + 13x - 56\), our objective is to find its roots by factoring. The roots of a quadratic equation are the values of \(x\) that make it equal to zero. Factoring the equation involves breaking it down into simpler multiplicative components, allowing for straightforward root calculation.Understanding the nature of a quadratic equation helps in selecting the best method to solve it and reach the required solution efficiently.

coefficient identification

Identifying coefficients is the crucial first step in solving a quadratic equation. It refers to recognizing the specific numbers in front of the variables and the constant term.In \(ax^2 + bx + c\), the coefficients are:

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

For the equation \(30x^2 + 13x - 56\):

• \(a = 30\)
• \(b = 13\)
• \(c = -56\)

Knowing these coefficients allows us to apply methodologies like the AC method and factoring by grouping effectively. It simplifies the complex-looking equation into understandable steps, kick-starting the entire process of factoring and solving.