Problem 18 - 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. \(25+4 x^{2}=-20 x\)

Step-by-Step Solution

Answer

The discriminant is 0, indicating one repeated real root, which is \( x = -\frac{5}{2} \).

Step 1: Rearrange the Equation

First, we need to rearrange the given equation into the standard form of a quadratic equation, which is \[ ax^2 + bx + c = 0 \]Starting with the given equation:\[ 25 + 4x^2 = -20x \]Move all terms to one side to form:\[ 4x^2 + 20x + 25 = 0 \]

Step 2: Identify the Coefficients

Identify the coefficients from the rearranged quadratic equation \[ 4x^2 + 20x + 25 = 0 \],where \[ a = 4,\ b = 20,\ c = 25 \].

Step 3: Calculate the Discriminant

The discriminant of a quadratic equation \( ax^2 + bx + c = 0 \) is calculated as \[ D = b^2 - 4ac \].Substitute the identified values:\[ D = 20^2 - 4 \times 4 \times 25 \]\[ D = 400 - 400 \]\[ D = 0 \]So, the value of the discriminant is 0.

Step 4: Describe the Number and Type of Roots

The value of the discriminant helps in determining the number and type of roots:- If \( D > 0 \): Two distinct real roots.- If \( D = 0 \): One real root (a repeated root).- If \( D < 0 \): Two complex roots.Since \( D = 0 \), there is one real and repeated root.

Step 5: Use the Quadratic Formula

The quadratic formula for finding the roots is given by:\[ x = \frac{-b \pm \sqrt{D}}{2a} \]Substitute the values of \( b = 20 \), \( a = 4 \), and \( D = 0 \):\[ x = \frac{-20 \pm \sqrt{0}}{8} \]\[ x = \frac{-20}{8} \]\[ x = -\frac{5}{2} \]So the exact solution is \( x = -\frac{5}{2} \).

Key Concepts

Understanding the Discriminant

The discriminant is a key component of a quadratic equation, and it's given by the formula \( D = b^2 - 4ac \). This value helps us understand what type of roots a quadratic equation will have. Here’s how:

• When \( D > 0 \), the quadratic equation has two distinct real roots.
• When \( D = 0 \), there is exactly one real root, often called a repeated or double root.
• When \( D < 0 \), the quadratic equation has no real roots; instead, it has two complex roots.

In our example equation, \( 4x^2 + 20x + 25 = 0 \), the discriminant is calculated as \( D = 20^2 - 4 \times 4 \times 25 = 0 \). Since the discriminant is 0, we know immediately that the equation has a single real root. This repetition means that the parabola just touches the x-axis at one point, reflecting the unique nature of this root.

Applying the Quadratic Formula

The quadratic formula is a universal tool for solving any quadratic equation of the form \( ax^2 + bx + c = 0 \). It is expressed as:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]
Using this formula, we can find the roots of the quadratic equation by substituting the values of \( a \), \( b \), and the discriminant \( D \).
In the given equation with \( a = 4 \), \( b = 20 \), and \( D = 0 \), the formula simplifies into:
\[ x = \frac{-20 \pm \sqrt{0}}{2 \times 4} \]
\( \sqrt{0} \) equals zero, which simplifies our expression to:
\[ x = \frac{-20}{8} \]
This yields the root \( x = -\frac{5}{2} \). Therefore, the quadratic formula confirms that there is one real, repeated root at this same value. By always having the quadratic formula handy, you can solve any quadratic equation step by step confidently.

Interpreting Real Roots

Real roots are solutions to a quadratic equation that can be plotted on the x-axis of a graph. They are actual numbers that solve the equation \( ax^2 + bx + c = 0 \). Understanding the nature of these roots is crucial to interpreting the solution:

• A real root means that the parabola formed by the quadratic equation intersects the x-axis.
• One repeated real root signifies that the vertex of the parabola lies on the x-axis exactly at that root.
• Two distinct real roots imply that the parabola crosses the x-axis at two separate points.

In the example provided, since \( D = 0 \), we have one real root \( x = -\frac{5}{2} \). This singular point indicates where the parabola touches the x-axis once, representing a precise balance where the expression equals zero without crossing it. Being able to identify and interpret these real roots can greatly enhance the understanding of quadratic relationships.