Problem 58 - Elementary Algebra

Question

\(m^{2}+24 m+150=0\)

Step-by-Step Solution

Answer

The roots of the equation are \(-12 + i\sqrt{6}\) and \(-12 - i\sqrt{6}\).

Step 1 - Identify the coefficients

To solve the quadratic equation, identify the coefficients from the equation: \(m^{2} + 24m + 150 = 0\). Here, \(a = 1\), \(b = 24\), and \(c = 150\).

Step 2 - Use the quadratic formula

Apply the quadratic formula: \[m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. Substitute \(a = 1\), \(b = 24\), and \(c = 150\).

Step 3 - Calculate the discriminant

Calculate the discriminant: \(b^2 - 4ac = 24^2 - 4(1)(150) = 576 - 600 = -24\).

Step 4 - Determine the nature of the roots

Since the discriminant is negative (\(-24\)), the equation has two complex roots. They can be found using the formula.

Step 5 - Solve for the roots

Calculate the roots: \[m = \frac{-24 \pm \sqrt{-24}}{2(1)} = \frac{-24 \pm i \sqrt{24}}{2} = -12 \pm i \sqrt{6}\]

Key Concepts

Quadratic Formula

The quadratic formula is a tool used to find the roots of a quadratic equation. A quadratic equation is typically in the form:
\[ ax^2 + bx + c = 0 \]
Here, the quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Where:

• a is the coefficient of the quadratic term
• b is the coefficient of the linear term
• c is the constant term

When solving the equation using the quadratic formula, plug in the values of a, b, and c to compute the roots.

Discriminant

The discriminant is a component that helps us determine the nature of the roots of a quadratic equation. It is given by:
\[ \Delta = b^2 - 4ac \]
Where

• \(b^2\)
• \(4ac\)

The value of the discriminant provides crucial information:

• If \(\Delta > 0\), there are two distinct real roots.
• If \(\Delta = 0\), there is one real root (a repeated root).
• If \(\Delta < 0\), there are two complex roots.

Complex Roots

Complex roots arise when the discriminant of a quadratic equation is negative. In such a case:
\[ \Delta < 0 \]
The roots are no longer real numbers. Instead, they include imaginary numbers. Imaginary numbers are expressed as:

• \(i\), where \(i\) is the imaginary unit, and \(i^2 = -1\)

For the given quadratic equation:

• \(m^2 + 24m + 150 = 0\)
• The discriminant is \(b^2 - 4ac\)
• 24\(24^2 - 4 \times 1 \times 150 = -24\)
• 
 Thus, the equation has complex roots: \[ m = \frac{-24 \pm \sqrt{-24}}{2(1)} \]
Solve for the imaginary part to get: \[ m = -12 \pm i \sqrt{6} \]
Complex roots often appear in conjugate pairs such as:
• \(a + bi\)
• \(a - bi\)

Polynomial Coefficients

In a polynomial equation like \(ax^2 + bx + c = 0\), the coefficients are the numerical values that multiply the variables:

• is called the quadratic coefficient (multiplies \(x^2\))
• is the linear coefficient (multiplies \(x\))
• is the constant coefficient (an independent term)

For example, in the equation \(m^2 + 24m + 150 = 0\):

• a = 1
• b = 24
• c = 150

These coefficients are essential when using methods like factoring, completing the square, or applying the quadratic formula to solve the equation. Identifying the coefficients correctly is the first step in these methods and ensures accurate solutions.