Problem 57 - Precalculus : Building Concepts and Connections

Question

Compute the zeros of the quadratic function. $$f(x)=-2 x^{2}-2 x+11$$

Step-by-Step Solution

Answer

The zeros of the quadratic function \(f(x)=-2 x^{2}-2 x+11\) are \(x_{1} = 4.29583\) and \(x_{2} = -5.29583\).

Step 1: Identify Coefficients a, b and c

In the quadratic function \(f(x)=-2x^{2}-2x+11\), identify \(a=-2\), \(b=-2\), and \(c=11\).

Step 2: Compute Determinant Value

Calculate the determinant \(\Delta\) which is \(b^2-4ac\). That gives us \((-2)^2 - 4*(-2)*11 = 4 + 88 = 92\).

Step 3: Apply Quadratic Formula

Apply the quadratic formula to find roots: \(x = \frac{-b \pm \sqrt{\Delta}}{2a} \). So, \(x = \frac{2 \pm \sqrt{92}}{2*-2}\). Simplifying this, we get the roots \(x = -0.5 \pm 4.79583\). Therefore, we have two real roots: \(x_{1} = -0.5 + 4.79583 = 4.29583\) and \(x_{2} = -0.5 - 4.79583 = -5.29583\).

Key Concepts

Quadratic Formula

The quadratic formula is a pivotal tool used to find the zeros of a quadratic function, which are the points where the graph of the function crosses or touches the x-axis. It is particularly helpful when factoring is difficult or impossible. The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \]
Here, the symbols 'a', 'b', and 'c' represent the coefficients of the quadratic equation in standard form, which is \( ax^2 + bx + c = 0 \). The expression under the square root, \( b^2-4ac \), is known as the determinant or the discriminant. The quadratic formula allows us to calculate the roots, or solutions, by substituting our coefficients into this formula. Each solution represents a point at which the parabola defined by the quadratic equation will intersect the x-axis.
For instance, in the exercise, by applying the quadratic formula with our coefficients a = -2, b = -2, and c = 11, we are able to determine the roots of the function, signifying the zeros of the parabola. This formula is versatile and can be applied to any quadratic function to find its zeros.

Determinant of a Quadratic Equation

The determinant, often represented by the Greek letter Delta (\(\Delta\)), is a component of the quadratic formula that informs us about the nature of the roots of a quadratic equation without actually calculating them. It is calculated using the coefficients of the quadratic equation as \(\Delta = b^2 - 4ac\).
The value of the determinant can lead to three different scenarios:

• If \(\Delta > 0\), the quadratic equation has two distinct real roots.
• If \(\Delta = 0\), the quadratic equation has exactly one real root, also called a repeated or double root.
• If \(\Delta < 0\), the quadratic equation has no real roots but rather two complex roots.

In our example, the determinant \(\Delta = 92\), which is positive, indicating that the quadratic equation \( -2x^2 - 2x + 11 = 0 \) has two distinct real roots. These roots are also the zeros of the quadratic function, corroborating the presence of two points where the graph intersects the x-axis. The determinant thus provides a quick way to assess the type and nature of the roots of any quadratic equation. Understanding the determinant can help students predict the outcome even before using the quadratic formula.

Coefficients of a Quadratic Equation

Coefficients are the numerical factors that multiply the variables in an equation. In the standard form of a quadratic equation, \( ax^2 + bx + c = 0 \), 'a', 'b', and 'c' represent the coefficients.

Roles of Coefficients:

Each coefficient has a specific role:

• The coefficient 'a' affects the direction (upward or downward) and the width of the parabola.
• The coefficient 'b' influences the position and orientation of the parabola in relation to the y-axis and the vertex's horizontal position.
• The coefficient 'c' represents the y-intercept, where the parabola crosses the y-axis.

Identifying coefficients correctly is essential because they are directly used in the quadratic formula and determinant calculation. In our exercise, the coefficients \(a = -2, b = -2, c = 11\) determine the shape and position of the parabola, as well as enabling us to find the zeros using the quadratic formula. Understanding how these coefficients affect the graph of the quadratic function can enhance a student's comprehension of how quadratic equations represent certain parabolic shapes on a graph. Properly identifying the coefficients is a crucial step to successfully applying the quadratic formula, computing the determinant, and graphing the quadratic function.