Problem 14 - 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. 49

Step-by-Step Solution

Answer

The roots are rational and unequal, and there are 2 x-intercepts.

Step 1: Understanding the Discriminant

The discriminant of a quadratic equation \( ax^2 + bx + c = 0 \) is given by \( D = b^2 - 4ac \). The value of \( D \) can classify the nature of the roots. Here, the discriminant is given as 49.

Step 2: Classifying Roots based on Discriminant

If \( D > 0 \) and is a perfect square, the roots are rational and unequal. Since 49 is greater than zero and is a perfect square (\(7^2\)), the roots for this quadratic equation are rational and unequal.

Step 3: Determining the Number of x-intercepts

The quadratic function graph corresponds to its roots. With \( D > 0 \), and rational unequal roots, the quadratic function will cross the x-axis at two points, meaning it has 2 x-intercepts.

Key Concepts

Roots of a Quadratic Equation

The roots of a quadratic equation are the solutions of the equation where it equals zero. In a standard quadratic equation, written as \( ax^2 + bx + c = 0 \), the roots can be found using the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]These roots tell us where the graph of the quadratic function intersects the x-axis. Using the discriminant \( D = b^2 - 4ac \), we can understand the nature of these roots.

• If \( D > 0 \), the equation has two distinct real roots.
• If \( D = 0 \), there is exactly one real root or a repeated root.
• If \( D < 0 \), the roots are complex and not real.

Understanding the discriminant helps us not only solve quadratic equations efficiently but also predict how its graph behaves.

Nature of Quadratic Roots

The nature of quadratic roots greatly depends on the value of the discriminant \( D \). It helps determine whether the solutions can be rational, irrational, or complex. Here's a closer look at each scenario:

• **Rational and Unequal Roots**: If \( D > 0 \) and \( D \) is a perfect square, the roots are rational (can be expressed as fractions) and different.
• **Rational and Equal Roots**: If \( D = 0 \), there is just one root, repeated. This means both roots have the same value and can be expressed as a rational number.
• **Irrational and Unequal Roots**: If \( D > 0 \) but not a perfect square, the roots are irrational (cannot be expressed exactly by fractions) and different.
• **Complex or Not Real Roots**: If \( D < 0 \), there are no real roots because the square root of a negative number is imaginary.

In our given exercise, since \( D = 49 \) (a perfect square and greater than zero), the roots are rational and unequal.

X-Intercepts of Quadratic Function

The x-intercepts of a quadratic function are the points where the graph crosses the x-axis. These intercepts directly correspond to the roots of the quadratic equation, which occur when \( f(x) = 0 \). Understanding the discriminant thus becomes crucial in determining these points.

• **Two X-Intercepts**: If \( D > 0 \), the function intersects the x-axis at two separate points, corresponding to the two distinct real roots.
• **One X-Intercept**: If \( D = 0 \), the quadratic graph just touches the x-axis at exactly one point, indicating a double root.
• **No X-Intercepts**: If \( D < 0 \), the graph does not intersect the x-axis at all, meaning the roots are complex.

For our exercise, \( D = 49 \) results in two distinct rational roots, thus giving us two x-intercepts on the graph.