Problem 68 - Precalculus : Building Concepts and Connections

Question

For function of the form \(f(x)=\) \(a x^{2}+b x+c,\) find the discriminant, \(b^{2}-4 a c,\) and use it to determine the number of \(x\)-intercepts of the graph of \(f .\) Also determine the number of real solutions of the equation \(f(x)=0.\) $$f(x)=3 x^{2}-4 x+4$$

Step-by-Step Solution

Answer

The function \(f(x)=3x^{2}-4x+4\) does not have any real solutions or x-intercepts as the discriminant is less than zero.

Step 1: Identify the values of a, b, c

From the given function f(x)=3x^{2}-4x+4, the values that can be seen are a = 3, b = -4, and c = 4.

Step 2: Substitute the values of a, b, c into the discriminant formula

Substitute the values of a, b, and, c into the discriminant formula to get \(-4^{2}-4*3*4\).

Step 3: Calculate the discriminant

After performing the calculations, the discriminant D equals \(16 - 48 = -32\).

Step 4: Conclude the number of x-intercepts and real solutions

Since the discriminant is less than zero, it can be concluded that the function \(f(x)=3x^{2}-4x+4\) has no real solutions and therefore, no x-intercepts.

Key Concepts

Quadratic Equations

A quadratic equation is a second-degree polynomial equation in a single variable x with a non-zero coefficient of the x² term. It has the standard form f(x) = ax2 + bx + c where a, b, and c are constants, and a ≠ 0. These equations are fundamental in algebra and appear in various applications across different fields, such as physics, engineering, and economics.

The solutions to a quadratic equation are the values of x that make the equation equal to zero. These solutions can be real or complex, and their nature is determined by a particular part of the equation known as the discriminant.

Number of Real Solutions

The number of real solutions to a quadratic equation can be found using the discriminant, which is the part under the square root in the quadratic formula. The discriminant is given by D = b² - 4ac. If D > 0, the quadratic equation has two distinct real solutions. If D = 0, there is exactly one real solution, also known as a repeated or double root. However, if D < 0, there are no real solutions, and instead, there are two distinct complex solutions.

Understanding the discriminant provides insight into the nature of the solutions without actually solving the equation. In our exercise, since the discriminant is -32, which is less than zero, we can determine there are no real solutions to the given quadratic equation.

X-Intercepts of a Parabola

The x-intercepts of a parabola, which are the points where the graph crosses the x-axis, correspond to the real solutions of the quadratic equation f(x) = 0. In other words, they are the roots of the equation. A parabola can have at most two x-intercepts, one intercept, or none, depending on the discriminant value.

If the discriminant is positive, the parabola crosses the x-axis at two points, revealing two distinct x-intercepts. If the discriminant is zero, the parabola touches the x-axis at just one point, indicating a single x-intercept. Finally, if the discriminant is negative, the parabola does not cross the x-axis at all, implying that there are no x-intercepts. This insight is crucial for graphing the parabola and understanding its intersection with the x-axis.

Quadratic Formula

The quadratic formula is a powerful tool used to find the solutions of a quadratic equation. The formula is x = (-b ± √D) / (2a), where D is the discriminant (b² - 4ac). The symbol '±' indicates there can be two solutions, one obtained by using the plus sign and the other with the minus sign, corresponding to the respective x-intercepts of the parabola.

This formula provides the exact solutions for any quadratic equation and is derived by completing the square of the standard quadratic equation. It is an essential concept in algebra because it guarantees a method to find all possible solutions of a quadratic equation, whether they are real or complex.