Problem 5 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Solve the equation by factoring, if required: $$ x^{2}+x-12=0 $$

Step-by-Step Solution

Answer

The given quadratic equation, \(x^2 + x - 12 = 0\), can be factored as \((x + 4)(x - 3) = 0\). Solving for x, we get two solutions: \(x = -4\) and \(x = 3\).

Step 1: Write down the equation

First, let's write down the given equation: \(x^2 + x - 12 = 0\)

Step 2: Factoring

Now we need to factor the quadratic equation. In this case, we need to find two numbers that multiply to -12 and add up to +1 (coefficient of 'x'). After testing different pairs we find that: +4 * -3 = -12 and +4 + -3 = +1 So, the equation can be factored as: \((x + 4)(x - 3) = 0\)

Step 3: Solve for x

Now, we have the factored equation. We can solve for x by setting each factor to zero: \(x + 4 = 0\) \(x - 3 = 0\) Solve each equation separately: For \(x + 4 = 0\), Subtract 4 from both sides: \(x = -4\) For \(x - 3 = 0\), Add 3 to both sides: \(x = 3\) Thus, the solutions to the given equation are x = -4 and x = 3.

Key Concepts

Factoring Quadratic Equations

Factoring quadratic equations is often the first method we try when solving these types of problems. It's because it allows us to break down the equation into simpler expressions that we can work with more easily. In a quadratic equation like \( ax^2 + bx + c = 0 \), we aim to find two numbers that multiply to the constant term \( c \) and add up to the coefficient of the linear term \( b \).

• This process often involves trial and error, as you test different number pairs to find the right combination.
• Once the correct pair is found, it allows us to rewrite the quadratic as a product of two binomials.
• These binomials, when multiplied, recreate the original quadratic equation.

In our example \( x^2 + x - 12 = 0 \), finding numbers that multiply to -12 and add to 1 led us to the pair +4 and -3. Thus, we can factor the equation as \((x + 4)(x - 3) = 0\). This expression is far simpler to solve.

Roots of Quadratic Equations

The roots of a quadratic equation, also known as its solutions or zeros, are the values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \). Once a quadratic is factored, finding the roots becomes a matter of setting each binomial factor equal to zero.

• This approach exploits the zero-product property, which states that if a product of factors equals zero, at least one of the factors must be zero.
• For instance, if you have a pair of factors like \((x + 4)(x - 3) = 0\), setting each factor equal to zero gives us two separate equations.
• Solve these equations individually to find \( x = -4 \) and \( x = 3 \) in our example.

By determining these values, you discover where the parabola intersect the x-axis. These intersections are vital, as they provide real-world solutions in many applications, from physics to finance.

Equation Solving Techniques

Quadratic equations can be solved using various techniques, and understanding them helps in choosing the most efficient method for a given problem.

• **Factoring**: This is effective when a quadratic can be easily rewritten as a product of binomials. It’s typically the simplest method.
• **Completing the Square**: This technique transforms the equation into a perfect square trinomial, allowing for easy solving by taking square roots.
• **Quadratic Formula**: This is a universal method that applies to any quadratic equation, using the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

Each method has its pros and cons. Factoring is quick but not always possible. The quadratic formula is always applicable but might be complex. Completing the square offers clarity but can be cumbersome.
In our initial problem, factoring was straightforward and allowed us to quickly find solutions: \( x = -4 \) and \( x = 3 \). Knowing how to choose and apply the right technique ensures efficiency and success in solving quadratic equations.