Problem 9 - 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. $$3 x^{2}+4 x=2$$

Step-by-Step Solution

Answer

The equation has two real solutions: \(x = \frac{-2 + \sqrt{10}}{3}\) and \(x = \frac{-2 - \sqrt{10}}{3}\).

Step 1: Set Equation to Standard Form

First, we need to arrange the equation in the standard quadratic form, which is \(ax^2 + bx + c = 0\). The given equation is \(3x^2 + 4x = 2\). Subtract 2 from both sides to get \(3x^2 + 4x - 2 = 0\).

Step 2: Identify Coefficients

Identify the coefficients in the quadratic equation \(ax^2 + bx + c = 0\). For the equation \(3x^2 + 4x - 2 = 0\), we have \(a = 3\), \(b = 4\), and \(c = -2\).

Step 3: Calculate the Discriminant

The discriminant \(D\) of a quadratic equation \(ax^2 + bx + c = 0\) is calculated using the formula \(D = b^2 - 4ac\). Substitute the values of \(a\), \(b\), and \(c\) into this formula: \(D = 4^2 - 4 \times 3 \times (-2)\). Calculate this to find \(D = 16 + 24 = 40\).

Step 4: Interpret the Discriminant

A discriminant \(D > 0\) indicates two distinct real solutions. Since \(D = 40\), which is greater than zero, the equation has two real solutions.

Step 5: Use the Quadratic Formula

To find the roots of the equation, use the quadratic formula: \(x = \frac{-b \pm \sqrt{D}}{2a}\). Substitute the values of \(b\), \(D\), and \(a\) into the formula: \(x = \frac{-4 \pm \sqrt{40}}{2 \times 3}\).

Step 6: Simplify the Solutions

First, simplify \(\sqrt{40}\) to \(2\sqrt{10}\). The expression becomes \(x = \frac{-4 \pm 2\sqrt{10}}{6}\). Simplify further by dividing each term in the numerator by 2 to get \(x = \frac{-2 \pm \sqrt{10}}{3}\). Thus, the solutions are \(x = \frac{-2 + \sqrt{10}}{3}\) and \(x = \frac{-2 - \sqrt{10}}{3}\).

Key Concepts

Discriminant

The discriminant is a key component when working with quadratic equations. It helps determine the nature of the roots without actually solving the equation. You can find the discriminant using the formula:

• \(D = b^2 - 4ac\)

The values of \(a\), \(b\), and \(c\) are coefficients from the standard form of a quadratic equation \(ax^2 + bx + c = 0\).
By calculating \(D\), you can tell:

• If \(D > 0\), there are two distinct real solutions.
• If \(D = 0\), there is one real solution (or a repeated solution).
• If \(D < 0\), there are two nonreal complex solutions.

In our exercise, we found \(D = 40\), indicating two distinct real solutions. Always remember, the sign and value of the discriminant are central to predicting the solution types.

Real Solutions

Real solutions related to quadratic equations refer to the actual values of \(x\) that can be plotted on a real number line. Depending on the discriminant, finding real solutions is possible in certain cases:

• When \(D > 0\), solutions are real and different.
• When \(D = 0\), you have one real solution with a multiplicity of two.

These solutions arise from the fact that a parabola (the graph of a quadratic equation) can intersect the x-axis. This intersection points represent the real solutions.
In the example of the quadratic we solved, we ended up with two different real solutions at \(x = \frac{-2 + \sqrt{10}}{3}\) and \(x = \frac{-2 - \sqrt{10}}{3}\). This means that, graphically, the parabola intersects the x-axis at these two points.

Quadratic Formula

The quadratic formula is a universal tool for finding the solutions to any quadratic equation of the form \(ax^2 + bx + c = 0\). It's given by:

• \(x = \frac{-b \pm \sqrt{D}}{2a}\)

where \(D\) is the discriminant \(b^2 - 4ac\). This powerful formula provides both real and complex solutions depending on the value of \(D\).
To use the quadratic formula, you simply need to substitute the values of \(a\), \(b\), and \(D\) into it. In our example, with \(a = 3\), \(b = 4\), and \(D = 40\), the formula leads to:

• \(x = \frac{-4 \pm \sqrt{40}}{6}\)

After simplifying, we found the roots as \(x = \frac{-2 + \sqrt{10}}{3}\) and \(x = \frac{-2 - \sqrt{10}}{3}\).
The quadratic formula is your go-to method for handling any quadratic equation, providing clear solutions based on the discriminant.