Problem 87 - Algebra for College Students

Question

Find all real or imaginary solutions to each equation. Use the method of your choice. $$m^{2}+2 m-24=0$$

Step-by-Step Solution

Answer

The solutions are \(m = 4\) and \(m = -6\).

Step 1: Identify the equation type

This is a quadratic equation of the form \[ax^2 + bx + c = 0\]. Here, \(a = 1\), \(b = 2\), and \(c = -24\).

Step 2: Choose the method

The quadratic formula can solve this: \[m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\].

Step 3: Calculate the discriminant

The discriminant is given by \[b^2 - 4ac\]. Substitute the values of \(a\), \(b\), and \(c\): \[2^2 - 4(1)(-24)= 4 + 96 = 100\].

Step 4: Solve for m using the quadratic formula

Substitute the values of \(a\), \(b\), and the discriminant into the quadratic formula: \[m = \frac{-2 \pm \sqrt{100}}{2(1)} = \frac{-2 \pm 10}{2}\].

Step 5: Find the solutions

Simplify the equations: \[m = \frac{-2 + 10}{2} = 4\] and \[m = \frac{-2 - 10}{2} = -6\]. Therefore, the solutions are: \(m = 4\) and \(m = -6\).

Key Concepts

solving quadratic equations

Solving quadratic equations can initially seem challenging, but it becomes much easier once you understand the core concepts. A quadratic equation is any equation of the form ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The equation you are solving here is m^2 + 2m - 24 = 0, which fits this format.
There are several methods to solve quadratic equations, including:

• Factorization
• Completing the square
• Using the quadratic formula

In this example, you used the quadratic formula. We'll delve into how this method works, as it is widely applicable.

discriminant

The discriminant is a key part of the quadratic formula. It helps determine the nature of the roots of the equation. The discriminant (D) is defined as: \[D = b^2 - 4ac\]
By substituting a=1, b=2, and c=-24 into the formula, you get: 2^2 - 4(1)(-24) = 4 + 96 = 100.
The value of the discriminant gives you important information:

• If D > 0, there are two distinct real roots.
• If D = 0, there is exactly one real root.
• If D < 0, there are two complex roots.

Here, D = 100, which indicates two distinct real roots. This is significant because it informs you that the equation has two real solutions.

quadratic formula

The quadratic formula is a universally applicable method for finding the roots of any quadratic equation. The formula is written as: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Using this formula requires calculating the discriminant first. For the given equation, you identified that a=1, b=2, and c=-24, and found the discriminant to be 100.
Plugging these values into the quadratic formula, you get: \[m = \frac{-2 \pm \sqrt{100}}{2}\]
This simplifies to: \[m = \frac{-2 \pm 10}{2}\]
Solving this gives you two results:

• \(m = \frac{-2 + 10}{2} = 4\)
• \(m = \frac{-2 - 10}{2} = -6\)

Therefore, the solutions to the quadratic equation are m = 4 and m = -6. The quadratic formula is incredibly useful because it provides a direct way to solve quadratic equations, even when factorization is difficult or impossible.