Problem 11 - Algebra 2 and Trigonometry

Question

In \(9-14\) : a. For each given value of the discriminant of a quadratic equation with rational coefficients, determine if the roots of the quadratic equation are \((1)\) rational and unequal, \((2)\) rational and equal, \((3)\) irrational and unequal, or \((4)\) not real numbers. b. Use your answer to part a to determine the number of \(x\) -intercepts of the graph of the corresponding quadratic function. 0

Step-by-Step Solution

Answer

Roots are rational and equal, with one x-intercept.

Step 1: Understand the Discriminant

The discriminant of a quadratic equation in the form \(ax^2 + bx + c = 0\) is given by \(b^2 - 4ac\). This value helps determine the nature of the roots of the quadratic equation.

Step 2: Evaluate the Discriminant's Value

We are given that the discriminant \(b^2 - 4ac = 0\). We will use this to determine the nature of the roots of the quadratic equation.

Step 3: Determine Root Nature Using the Discriminant

If the discriminant \(b^2 - 4ac = 0\), the quadratic equation has rational and equal roots. This means that the roots are repeated, and thus, there is only one distinct solution.

Step 4: Determine the Number of x-intercepts

Since the quadratic equation has rational and equal roots, its graph touches the x-axis at exactly one point. Therefore, there is one x-intercept for the graph of the quadratic function.

Key Concepts

Quadratic Equation

A quadratic equation is an expression of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\).
This type of equation represents a parabola when graphed on a coordinate plane. The highest power of \(x\) is 2, which is why it's called a "quadratic," derived from "quadrate," meaning square.
Quadratic equations are a key part of algebra and appear in various problem-solving scenarios in math and science.
To solve a quadratic equation, we often use the quadratic formula, **factorization**, or **completing the square**.

• Using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \]
• Ensuring factorization: express the equation as the product of linear factors.
• Completing the square: reformat the equation to a perfect square trinomial.

Each method is useful in different contexts and can reveal the quantity and type of solutions, or "roots."

Roots of Quadratic Equation

The roots of a quadratic equation are the solutions \(x\) that satisfy \(ax^2 + bx + c = 0\). These roots can be found using the quadratic formula, factoring, or by graphing the equation.
The roots indicate the points where the quadratic function intersects the x-axis, if at all. In solving the equation, the discriminant \(b^2 - 4ac\) plays a crucial role:

• If \(b^2 - 4ac > 0\), there are two distinct real roots.
• If \(b^2 - 4ac = 0\), the roots are real and equal.
• If \(b^2 - 4ac < 0\), the roots are not real.

The roots determine the characteristics of the parabola's graph and its position relative to the x-axis.

x-intercepts

x-intercepts are the points where a graph crosses or touches the x-axis. For a quadratic equation, these are precisely where the roots lie.
When the discriminant is evaluated, it helps to determine the existence and quantity of these intercepts:

• Two distinct x-intercepts if the discriminant is positive.
• One intercept if the discriminant is zero, as the graph just touches the x-axis.
• No x-intercepts if the discriminant is negative, indicating the parabola does not cross the x-axis at all.

Understanding x-intercepts is vital when interpreting the physical phenomena or datasets that quadratic equations might describe.

Nature of Roots

The **nature of roots** pertains to the type and number of solutions a quadratic equation has. This nature is dictated by the value of the discriminant.
Upon calculating the discriminant, we can categorize the roots:

• **Rational and unequal**: The discriminant is positive and a perfect square.
• **Rational and equal**: The discriminant is exactly zero, indicating a single solution repeated.
• **Irrational and unequal**: The discriminant is positive but not a perfect square.
• **Not real numbers**: The discriminant is negative, suggesting complex solutions.

By exploring the nature of roots, students can understand the behavior of quadratic functions and their respective graphs more thoroughly, aiding in better analysis and problem-solving.