Problem 83 - Precalculus : Building Concepts and Connections

Question

Use the function \(f(x)=a x^{2}+2 x+1\) where a is a real number. For what value(s) of \(a\) will \(f\) have one real zero?

Step-by-Step Solution

Answer

The function \(f(x) = ax^2 + 2x + 1\) will have one real zero when \(a = 1\).

Step 1: Identify the coefficients

From the function \(f(x) = ax^2 + 2x + 1\), we can identify the coefficients for the quadratic formula \(-b ± \sqrt{b^2 - 4ac}\) by comparison, where the quadratic equation is in the form \(ax^2 + bx + c = 0\). The coefficients are \(a = a\) (from the \(f(x)\) equation), \(b = 2\), and \(c = 1\).

Step 2: Calculate the discriminant

The discriminant of a quadratic function is given by the formula \(D = b^2 - 4ac\). Substituting the identified values, \(D = 2^2 - 4a * 1 = 4 - 4a\).

Step 3: Find the zero discriminant

A quadratic equation has one real solution when its discriminant is zero. So we equate the discriminant to zero and solve for \(a\), as such: \(0 = 4 - 4a\).

Step 4: Solve for a

To find \(a\), we solve the equation \(0 = 4 - 4a\) to yield \(a = 1\).

Key Concepts

Real Zeros

Real zeros are the values of \(x\) where a quadratic function intersects the x-axis of a graph. These zeros are essentially the solutions to the quadratic equation.
When a quadratic function is expressed in the form \(f(x) = ax^2 + bx + c\), finding real zeros means solving for the values of \(x\) such that \(f(x) = 0\).
In practical terms, solving \(ax^2 + bx + c = 0\) can result in:

• Two distinct real zeros when the discriminant is positive.
• One real zero (a repeated root) when the discriminant is zero.
• No real zeros when the discriminant is negative.

In our original exercise, we determined the condition for having exactly one real zero, which occurs when the discriminant \(D\) is zero. This results in the quadratic function just touching the x-axis rather than crossing it.

Discriminant

The discriminant is a crucial concept in determining the nature of the solutions for a quadratic equation. It is derived from the standard form of a quadratic equation \(ax^2 + bx + c = 0\) and is given by the formula:
\(D = b^2 - 4ac\).
The discriminant tells us:

• If \(D > 0\), the equation has two distinct real zeros.
• If \(D = 0\), there is one real zero, indicating the function has a perfect square root.
• If \(D < 0\), the equation has no real zeros, meaning the roots are complex or imaginary.

In our exercise, we solved for \(a\) when the discriminant was set to zero, \(D = 4 - 4a = 0\), revealing that the function has one real zero when \(a = 1\). This shows how critical the discriminant is for understanding the behavior of quadratic functions.

Quadratic Formula

The quadratic formula provides a consistent method for finding the zeros of any quadratic equation, given by \(ax^2 + bx + c = 0\). The formula is:
\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
This formula derives solutions by encompassing the discriminant \(b^2 - 4ac\), which is under the square root, indicating that the discriminant directly affects the number and type of solutions.
The 'plus-minus' sign denotes there are typically two solutions, except when the discriminant is zero, where both solutions converge to the same value.
This shows clearly in the exercise where the calculated \(D = 0\). Applying this condition to the quadratic formula results in one real zero. The formula thus makes it possible to solve any quadratic equation systematically, ensuring an understanding of the relationship between its coefficients and the nature of its roots.