Problem 10 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Solve the equation by factoring, if required: $$ \frac{1}{2} a^{2}+a-12=0 $$

Step-by-Step Solution

Answer

The short answer based on the provided step-by-step solution is: To solve the quadratic equation \(\frac{1}{2} a^{2} + a - 12 = 0\), multiply all terms by the least common denominator (LCD = 2), factor the resulting equation (\(a^{2} + 2a - 24 = 0\)), and then find the solutions for 'a' by setting each factor equal to zero. The solutions are \(a = -6\) and \(a = 4\).

Step 1: Eliminate the fraction

To eliminate the fraction, multiply all the terms by the least common denominator (LCD) which is 2. \[\begin{aligned} 2\left(\frac{1}{2}a^{2}+a-12\right)&= 2 (0) \\ \\ a^{2}+ 2a - 24 &= 0 \end{aligned}\] Now we have the quadratic equation as \(a^{2} + 2a - 24 = 0\).

Step 2: Factor the quadratic equation

Now, we will factor the quadratic equation: \((a+6)(a-4)=0\) We can see that the quadratic equation has been factored into two expressions multiplied together.

Step 3: Set each factor equal to zero and solve for 'a'

Next, we set each of the factors equal to zero and solve for 'a': 1. \(a+6=0\): Subtract 6 from both sides: \(a=-6\) 2. \(a-4=0\): Add 4 to both sides: \(a=4\) The solutions for the quadratic equation are \(a=-6\) and \(a=4\).

Key Concepts

Factoring Quadratics

Factoring quadratics is a method used for solving quadratic equations. It involves finding two binomials that multiply together to give the original quadratic expression. In our example, the given quadratic equation is:\[ a^2 + 2a - 24 = 0 \]This is a standard quadratic equation where we need to rewrite it in the form of \( ax^2 + bx + c = 0 \). Here, the task is to express \( a^2 + 2a - 24 \) as a product of two binomials.The process involves the following steps:

• First, find two numbers that multiply to the constant term, \(-24\), and add up to the middle term's coefficient, \(2\).
• After testing, the numbers \(6\) and \(-4\) are found to satisfy both conditions.
• Thus, the quadratic can be factored into \((a + 6)(a - 4) = 0\).

This method is effective as it reduces the quadratic expression into simpler terms that are easier to solve.

Solving Equations

Once a quadratic equation is factored into binomials, the next step is solving for the unknown variable. In our example, the equation was factored into:\( (a + 6)(a - 4) = 0 \).To find the solutions for \(a\), you apply the zero product property. This property states that if the product of two expressions is zero, then at least one of the expressions should be zero. This gives us two simpler equations to solve:

• \( a + 6 = 0 \)
• \( a - 4 = 0 \)

Solving these gives:

• For \( a + 6 = 0 \), subtract 6 from both sides to get \( a = -6 \).
• For \( a - 4 = 0 \), add 4 to both sides to find \( a = 4 \).

Hence, the solutions to the quadratic equation are \( a = -6 \) and \( a = 4 \). This simple set of actions transforms complex expressions into manageable data points, which provides clear solutions.

Algebraic Techniques

Algebraic techniques are foundational tools that are frequently used in solving mathematical problems, such as quadratic equations. These techniques include operations like

• Multiplying through by the least common denominator to clear fractions.
• Applying the distributive property to expand or factor expressions.
• Using the zero product property to solve equations quickly.

In the given problem, multiplying by the least common denominator (LCD) of 2 was our first step to clear the fraction in:\[ \frac{1}{2}a^2 + a - 12 = 0 \]This step allows us to work with integer values, simplifying the process of factoring.Another essential technique is the ability to spot patterns or relationships in numbers, such as recognizing pairs of numbers that sum to the middle coefficient and multiply to the constant term. These algebraic strategies enhance precision and efficiency when factoring and solving quadratic equations.By integrating these techniques proficiently, tricky algebra problems become significantly more approachable and straightforward to navigate.