Problem 22 - Algebra 2

Question

Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula. \(2 x-5=-x^{2}\)

Step-by-Step Solution

Answer

The discriminant is 24; two distinct real roots: \(x = -1 + \sqrt{6}\) and \(x = -1 - \sqrt{6}\).

Step 1: Write the equation in standard form

The given equation is \(2x - 5 = -x^2\). To write it in standard form, move all terms to one side of the equation. Add \(x^2\) to both sides: \(x^2 + 2x - 5 = 0\).

Step 2: Identify coefficients for the quadratic formula

In the standard form equation \(ax^2 + bx + c = 0\), identify \(a = 1\), \(b = 2\), and \(c = -5\).

Step 3: Calculate the discriminant

The discriminant \(D\) of the quadratic equation \(ax^2 + bx + c = 0\) is given by \(D = b^2 - 4ac\). Substitute \(a = 1\), \(b = 2\), \(c = -5\) into the formula: \[D = 2^2 - 4 \times 1 \times (-5) = 4 + 20 = 24\].

Step 4: Describe the number and type of roots

Since the discriminant \(D = 24\) is greater than zero, the quadratic equation has two distinct real roots.

Step 5: Apply the Quadratic Formula

The Quadratic Formula is \(x = \frac{-b \pm \sqrt{D}}{2a}\). Substitute \(a = 1\), \(b = 2\), and \(D = 24\): \[x = \frac{-2 \pm \sqrt{24}}{2 \times 1}\].

Step 6: Simplify the solutions

Calculate \(\sqrt{24} = 2\sqrt{6}\). The solutions are \[x = \frac{-2 \pm 2\sqrt{6}}{2}\], which simplify to \(x = -1 \pm \sqrt{6}\). Thus, the exact solutions are \(x = -1 + \sqrt{6}\) and \(x = -1 - \sqrt{6}\).

Key Concepts

Discriminant

A discriminant is a special value that helps us determine the nature of the roots of a quadratic equation. When working with quadratic equations in the form of \(ax^2 + bx + c = 0\), finding the discriminant can give you insight into the roots without having to solve the entire equation. The discriminant, represented as \(D\), is calculated using the formula:

• \(D = b^2 - 4ac\)

Where \(a\), \(b\), and \(c\) are the coefficients from the quadratic equation. Calculating the discriminant will tell you:

• If \(D > 0\): Two distinct real roots exist.
• If \(D = 0\): One real root exists, or the roots are considered as a repeated root.
• If \(D < 0\): No real roots exist, but there are two complex roots.

In our example, the discriminant was \(24\), which is greater than zero, indicating two distinct real roots.

Quadratic Formula

The Quadratic Formula is an incredibly useful tool for finding the roots of any quadratic equation. Given a quadratic equation in the standard form \(ax^2 + bx + c = 0\), the formula can be used to find the exact solutions. The Quadratic Formula is:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

This formula allows you to substitute your coefficients \(a\), \(b\), and \(c\) to find solutions for \(x\). The "\(\pm\)" symbol in the formula represents two potential solutions:

• \(x = \frac{-b + \sqrt{b^2 - 4ac}}{2a}\)
• \(x = \frac{-b - \sqrt{b^2 - 4ac}}{2a}\)

For our equation, substituting \(a = 1\), \(b = 2\), and \(c = -5\) into the formula gives solutions of \(x = -1 + \sqrt{6}\) and \(x = -1 - \sqrt{6}\). The formula not only provides a way to find the roots but also connects directly to the discriminant, since the term \(\sqrt{b^2 - 4ac}\) is the discriminant itself.

Real Roots

Real roots of a quadratic equation are the solutions for which the equation equals zero, and these solutions are visible on a graph as the x-intercepts where the curve crosses the x-axis. A quadratic equation can have different types of roots based on its discriminant value.

• If the discriminant is greater than zero, the equation has two distinct and real roots. This means the graph of the equation will intersect the x-axis at two separate points.
• If the discriminant is zero, there is one repeated real root, meaning the vertex of the parabola just touches the x-axis once.
• If the discriminant is less than zero, there are no real roots, and instead, the solutions are complex numbers. This means the graph does not cross the x-axis at all.

Understanding real roots is crucial for analyzing and graphing quadratic functions, as it provides immediate insight into the behavior and potential solutions of the equation.