Problem 1 - Algebra for College Students

Question

First use the discriminant to determine whether the equation has two nonreal complex solutions, one real solution with a multiplicity of two, or two real solutions. Then solve the equation. $$x^{2}+4 x-21=0$$

Step-by-Step Solution

Answer

The equation has two real solutions: 3 and -7.

Step 1: Identify the Coefficients

In the quadratic equation of the form \( ax^2 + bx + c = 0 \), identify the coefficients: \( a = 1 \), \( b = 4 \), and \( c = -21 \).

Step 2: Calculate the Discriminant

The discriminant \( D \) is calculated using the formula \( D = b^2 - 4ac \). Substitute the values: \( D = 4^2 - 4 \times 1 \times (-21) = 16 + 84 = 100 \).

Step 3: Determine the Nature of the Roots

Since the discriminant \( D = 100 \) is greater than zero, the quadratic equation has two distinct real solutions.

Step 4: Solve the Quadratic Equation

Use the quadratic formula \( x = \frac{-b \pm \sqrt{D}}{2a} \) to find the roots. Substitute the values: \( x = \frac{-4 \pm \sqrt{100}}{2 \times 1} = \frac{-4 \pm 10}{2} \).

Step 5: Calculate the Solutions

First, calculate \( x_1 = \frac{-4 + 10}{2} = \frac{6}{2} = 3 \). Then calculate \( x_2 = \frac{-4 - 10}{2} = \frac{-14}{2} = -7 \).

Key Concepts

The Discriminant

The discriminant is a crucial part of solving quadratic equations. It helps us understand the nature of the solutions without needing to solve the equation completely.

For any quadratic equation in the form of \( ax^2 + bx + c = 0 \), the discriminant \( D \) is given by the formula:

• \( D = b^2 - 4ac \)

The value of the discriminant tells us if the roots of the equation are real or complex. Here's how to interpret it:

• If \( D > 0 \), there are two distinct real solutions.
• If \( D = 0 \), there is exactly one real solution with a multiplicity of two (also known as a repeated or double root).
• If \( D < 0 \), the equation has two complex solutions, which are conjugates of each other.

In our current problem, the discriminant was calculated to be \( D = 100 \), which is greater than zero, indicating two distinct real solutions.

Real Solutions

Real solutions refer to the values of \( x \) that satisfy the quadratic equation and fit in the real number system.

When the discriminant of a quadratic equation is greater than zero, the equation will have two distinct real solutions. These solutions are the points where the quadratic function intersects the x-axis on a graph.

If the discriminant equals zero, the real solution is called a double or repeated root, meaning the parabola only touches the x-axis at one point.

In the case of our example equation with roots \( x = 3 \) and \( x = -7 \), there are two real solutions as calculated earlier.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations of any form, whether the solutions are real or complex.

It is expressed as:

• \( x = \frac{-b \pm \sqrt{D}}{2a} \)

Here, \( D \) represents the discriminant \( b^2 - 4ac \). The \( \pm \) symbol indicates that there are generally two solutions: one using the plus sign and the other using the minus sign.

In solving the equation \( x^2 + 4x - 21 = 0 \), we used this formula after finding \( D = 100 \).By substituting \( a = 1 \), \( b = 4 \), and \( D \), we calculated:

• \( x_1 = \frac{-4 + 10}{2} = 3 \)
• \( x_2 = \frac{-4 - 10}{2} = -7 \)

These are the two distinct real solutions, confirming our analysis of the discriminant.