Problem 2 - Algebra 2 and Trigonometry

Question

Rita said that when \(a, b,\) and \(c\) are real numbers, the roots of \(a x^{2}+b x+c=0\) are real numbers only when \(b^{2} \geq 4 a c .\) Do you agree with Rita? Explain why or why not.

Step-by-Step Solution

Answer

Yes, Rita is correct; real roots occur when \(b^2 \geq 4ac\).

Step 1: Understanding the Discriminant

The roots of a quadratic equation of the form \(a x^2 + bx + c = 0\) are determined using the discriminant formula, \(D = b^2 - 4ac\). This discriminant tells us the nature of the roots.

Step 2: Condition for Real Roots

For the roots of the quadratic equation to be real, the discriminant \(D\) must be greater than or equal to zero. That is, \(b^2 - 4ac \geq 0\). This ensures that the square root of the discriminant is a real number, allowing us to find two real roots using the quadratic formula.

Step 3: Interpreting Rita's Statement

Rita stated that the roots are real when \(b^2 \geq 4ac\). This condition is exactly what we found necessary for the discriminant to be non-negative, ensuring the roots are real. Thus, Rita's statement is correct.

Key Concepts

Discriminant

The discriminant is a special value found in quadratic equations that helps determine the nature of the roots of the equation. For a quadratic equation in the form of \(ax^2 + bx + c = 0\), the discriminant \(D\) is calculated using the formula \(D = b^2 - 4ac\).
The discriminant provides important information:

• If \(D > 0\), the equation has two distinct real roots.
• If \(D = 0\), the equation has exactly one real root, also known as a repeated or double root.
• If \(D < 0\), the equation does not have real roots; instead, the roots are complex or imaginary.

Understanding the discriminant is crucial in identifying whether an equation will yield real roots or not.

Real Roots

Real roots in a quadratic equation occur when the solutions to the equation can be plotted on the real number line and are not imaginary. The discriminant determines if and how many real roots an equation will have. For the quadratic equation \(ax^2 + bx + c = 0\), real roots are possible when the discriminant \(D = b^2 - 4ac\) is equal to or greater than zero.
- When \(D > 0\): The quadratic equation will cut the x-axis at two distinct points, meaning two separate real roots exist.- When \(D = 0\): The x-axis is touched at a single point indicating a repeated real root, which is also the vertex of the parabola represented by the equation.
Real roots are important as they represent practical solutions in real-world problems where imaginary numbers are not applicable.

Quadratic Formula

The quadratic formula is a powerful tool used to find the roots of a quadratic equation. For an equation \(ax^2 + bx + c = 0\), the roots are given by the formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula incorporates the discriminant \(b^2 - 4ac\), giving insight into both the nature and the value of the roots based on whether the discriminant is positive, zero, or negative.
When applying the quadratic formula:

• If \(D > 0\), the square root in the formula evaluates to a real number, allowing for the determination of two distinct real roots.
• If \(D = 0\), the expression simplifies to give one real root.
• If \(D < 0\), the square root yields an imaginary number, leading to complex roots.

The quadratic formula is essential for solving quadratic equations easily and efficiently, especially when factoring is complex or not possible.