Problem 37 - College Algebra and Calculus: An Applied Approach

Question

Algebraic and Graphical Approaches In Exercises \(31-46\), find all real zeros of the function algebraically. Then use a graphing utility to confirm your results. $$f(x)=2 x^{2}+4 x+6$$

Step-by-Step Solution

Answer

The function \(f(x) = 2x^{2} + 4x + 6\) has no real zeros since the discriminant of the quadratic equation is negative.

Step 1: Solve the quadratic equation

To find the zeros of the function, set \(f(x) = 0\). The given function is \(2x^{2} + 4x + 6 = 0\). To solve this equation for the variable \(x\), use the quadratic formula \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. Here, \(a = 2\), \(b = 4\), and \(c = 6\).

Step 2: Compute the Discriminant

Before proceeding to calculate the roots, find the value of the Discriminant. The Discriminant, \(D = b^2 - 4ac\), is used to determine the nature of the roots of a quadratic equation. If the Discriminant is less than zero the equation has no real solutions. In this case, \(D = 4^2 - 4*2*6 = 16 - 48 = -32\). Since the Discriminant is negative, there are no real zeros for the equation.

Step 3: Interpret The Result

Since the discriminant is negative (-32), this means the quadratic equation does not have any real roots. The given function \(f(x) = 2x^{2} + 4x + 6\) has no real zeros.

Key Concepts

Real Zeros

When we refer to the real zeros of a function, we are identifying the points where the graph of the function crosses or touches the x-axis. In simpler terms, these are the x-values for which the function equals zero. For polynomial functions, these zeros are also known as roots or solutions of the equation when set to zero.
Understanding that not every quadratic equation will have real zeros is important. The quadratic equation may have two, one, or no real zeros depending on its discriminant. Real zeros are critical in understanding the nature of graph intersections with the x-axis. In this exercise, since the discriminant was negative, the function had no real zeros.

Discriminant

The discriminant is a key part of the quadratic equation. It provides valuable information about the nature of the solutions of a quadratic equation without solving it entirely.
The formula for the discriminant \[D = b^2 - 4ac\] is derived from the quadratic formula and helps us predict the number and types of solutions the quadratic equation may have.

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If the discriminant is zero, it indicates one real root, essentially a repeated root where the graph just touches the x-axis.
• If the discriminant is negative, as in the exercise, there are no real roots, which implies the graph does not intersect the x-axis at any point.

In this exercise, the calculation revealed a discriminant of -32, confirming that there are no real zeros.

Quadratic Formula

The quadratic formula is a powerful tool used to solve quadratic equations of the form \[ax^2 + bx + c = 0\]. It is given by the formula \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\].
Here's how each component works:

• The symbol \(\pm\) in the formula means you will be doing two operations: one with addition and the other with subtraction.
• The part under the square root sign, known as the discriminant, gives us critical insight into the types of roots the equation will have.
• The terms \(-b\) represent how much the axis of symmetry shifts from the y-axis.

The quadratic formula can solve any quadratic equation, whether it has real roots, a single repeated real root, or complex roots. It is a universal tool, unlike methods such as factoring, which rely on simpler forms of quadratic equations. In this exercise, despite using the quadratic formula, the solutions resulted in complex numbers due to the negative discriminant, reaffirming no real zeros existed for the function.