Problem 11 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Solve the equation by factoring, if required: $$ 2 m^{2}+m=6 $$

Step-by-Step Solution

Answer

The short answer to the given quadratic equation \(2 m^{2}+m=6\) is \(m = -2\) and \(m = \frac{3}{2}\).

Step 1: Rewrite the equation in standard form

Subtract 6 from both sides of the equation to rewrite it in the standard form: \(2m^2 + m - 6 = 0\)

Step 2: Factor the equation

We can attempt to factor the quadratic equation using the "ac method". The "ac method" involves finding two numbers whose product is equal to the product 'ac' and whose sum is equal to the coefficient 'b'. In our case, \(a = 2\), \(b = 1\), and \(c = -6\). So, the product 'ac' = 2 * (-6) = -12 and the middle term's coefficient 'b' = 1. Find two numbers that multiply to -12 and add up to 1: 4 * (-3) = -12 and 4 + (-3) = 1 Now, rewrite the middle term using these two numbers and then factor by grouping: \(2m^2 + 4m - 3m - 6 = 0\) Factor by grouping: \(2m(m + 2) - 3(m + 2) = 0\) Now, we can factor out the common factor \((m + 2)\): \((m + 2)(2m - 3) = 0\)

Step 3: Solve for m

Now we have the factored form of the quadratic equation. Use the zero-product property to find the values for m: 1. \(m + 2 = 0\) 2. \(2m - 3 = 0\) Solving equations for m: 1. \(m + 2 = 0\): Subtract 2 from both sides: \(m = -2\) 2. \(2m - 3 = 0\): Add 3 to both sides, then divide by 2: \(m = \frac{3}{2}\) So, the solutions to the given quadratic equation are \(m = -2\) and \(m = \frac{3}{2}\).

Key Concepts

Factoring Quadratic Equations

When dealing with quadratic equations, a common approach involves rewriting the equation to make it easier to solve. Factoring is a popular strategy used to break down a quadratic equation into simpler parts. The goal is to express the quadratic in a product form, so that we can easily find its roots. Quadratic equations often follow this generic form: \( ax^2 + bx + c = 0 \), where \( a \), \( b \) and \( c \) are constants.

The first step in factoring is to ensure the equation is in "standard form", which means it is set to equal zero as shown: \( 2m^2 + m - 6 = 0 \). By setting it to zero, it allows us to eventually use strategies like the "zero-product property" to solve for the variable. Once in standard form, we look to decompose the equation into two binomials, like potential expressions \((x + p)(x + q)\).

The number of factors possible depends on the values of the coefficients \( a \), \( b \), and \( c \). Thus, factoring requires identifying component numbers that when multiplied together, give the original quadratic. If direct factoring is not possible right away, methods like the "AC method" can assist in breaking it down.

Zero-Product Property

The zero-product property is a fundamental concept in algebra, particularly when solving factored quadratic equations. This property states that if the product of two factors is zero, then at least one of the factors must be zero. This is the key principle used to determine the roots of an equation once it has been successfully factored.

Consider the example of the equation \((m + 2)(2m - 3) = 0\). Each factor stands a chance to be zero, giving us potential solutions for \( m \). To find these solutions, set each factor separately equal to zero:

• \( m + 2 = 0 \)
• \( 2m - 3 = 0 \)

Solving these equations yields \( m = -2 \) and \( m = \frac{3}{2} \). Thus, the zero-product property helps us derive these two possible values for \( m \).

By understanding this property, we see the power of factoring: it provides a direct path to the solutions of quadratic equations. This method applies consistently, allowing students to tackle a wide range of similar problems.

AC Method

The "AC Method" is a detailed and systematic way of factoring quadratic equations, especially when the coefficient \( a \) of \( ax^2 \) is not equal to 1. This method involves a few specific steps:

• Multiply the leading coefficient \( a \) by the constant term \( c \), yielding the product \( ac \).
• Find two numbers that multiply to \( ac \) and add up to the middle coefficient \( b \).

In our example, with equation \( 2m^2 + m - 6 = 0 \), we have \( a = 2 \), \( b = 1 \), and \( c = -6 \). Hence, the product \( ac \) is \(-12\). The goal is to find two numbers whose product is \(-12\) and whose sum is \(1\). These numbers turn out to be \(4\) and \(-3\).

Next, rewrite the middle term using these numbers: \( 2m^2 + 4m - 3m - 6 = 0 \). This allows you to group the terms in pairs and factor each group accordingly:

• First Group: \( 2m(m + 2) \)
• Second Group: \(-3(m + 2) \)

Finally, factor by grouping, taking out the common factor \((m + 2)\), which results in the equation \((m + 2)(2m - 3) = 0 \). Through these structured steps, the AC Method provides a clear path to factor and solve otherwise complex quadratic equations.