Problem 57 - Elementary Algebra

Question

\(m^{2}+24 m+167=0\)

Step-by-Step Solution

Answer

The solutions are \(m = -12 + i \sqrt{23}\) and \(m = -12 - i \sqrt{23}\).

Step 1 - Identify coefficients

The given quadratic equation is in the form of \(ax^2 + bx + c = 0\). Here, identify the coefficients: a = 1 (coefficient of \(m^2\)), b = 24 (coefficient of \(m\)), and c = 167 (constant term).

Step 2 - Apply the quadratic formula

To solve the quadratic equation, use the quadratic formula: \[m = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\]Substitute the values of a, b, and c: \[m = \frac{-24 \pm \sqrt{24^2-4(1)(167)}}{2(1)}\]

Step 3 - Calculate the discriminant

First, calculate the discriminant \(b^2 - 4ac\):\[24^2 - 4(1)(167) = 576 - 668 = -92\]Since the discriminant is negative, it implies the solutions are complex.

Step 4 - Simplify

Calculate the two solutions using the quadratic formula and the discriminant: \[m = \frac{-24 \pm \sqrt{-92}}{2}\]Since \(\sqrt{-92} = i \sqrt{92}\), rewrite the solutions as: \[m = \frac{-24 \pm i \sqrt{92}}{2}\]Further simplifying: \[m = -12 \pm i \sqrt{23}\]

Key Concepts

quadratic formula

A quadratic equation is any equation that can be written in the form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants.To solve such equations, we use the quadratic formula:\[m = \frac{-b \, \pm \,\sqrt{b^2-4ac}}{2a} \]This formula helps us find the values of \(m\) that satisfy the equation.

Let's break it down:

• \(b^2\) and \(-4ac\) terms make up the part under the square root, known as the **discriminant**.
• The whole expression under the square root sign is divided by \(2a\), so we need to be precise in our calculations.
• The plus-minus (\(\pm\)) symbol means we will get two different solutions, one with plus and one with minus.

In our case, for the equation \(m^2 + 24m + 167 = 0\), the formula will help us find the values of \(m\) by substituting our coefficients (\(a = 1\), \(b = 24\), and \(c = 167\)) into the formula.

complex solutions

Complex solutions arise when the discriminant (\(b^2 - 4ac\)), the part under the square root in the quadratic formula, is negative.
This means we're trying to take the square root of a negative number, which doesn't yield a real number.

Instead, we get complex numbers that include the imaginary unit \(i\), where \(i = \sqrt{-1}\). Here's how we deal with it in calculations:

• When the discriminant is negative (like our case where \(24^2 - 4(1)(167) = -92\)), it means you're dealing with imaginary numbers.
• We rewrite \(\sqrt{-92}\) as \(i \, \sqrt{92}\), adding the imaginary unit 'i'.
• This transforms our solution format into \(m = \frac{-24 \, \pm \, i \, \sqrt{92}}{2}\).

In the end, we can simplify further to \(m = -12 \, \pm \, i \, \sqrt{23}\). So, our complex solutions for \(m\) are: \(-12 + i \sqrt{23}\) and \(-12 - i \sqrt{23}\).

discriminant

The discriminant is a key part of the quadratic formula that tells us about the nature of the roots of a quadratic equation. It's given by the expression \(b^2 - 4ac\).

The value of the discriminant can indicate the type of solutions we get:

• If \(b^2 - 4ac > 0\), we get two distinct real solutions.
• If \(b^2 - 4ac = 0\), we get exactly one real solution (also known as a repeated root).
• If \(b^2 - 4ac < 0\), we get two complex solutions, meaning no real solutions are available.

For our equation \(m^2 + 24m + 167 = 0\), substituting \(b = 24\), \(a = 1\), and \(c = 167\), we found the discriminant to be \(-92\), which is less than zero. Therefore, we confirmed that the solutions are complex numbers. Understanding the discriminant helps predict the nature of the roots without fully solving the equation.