Problem 82 - 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 two real zeros?

Step-by-Step Solution

Answer

The quadratics function \(f(x) = ax^2 + 2x + 1\) will have two real zeros if the coefficient \(a\) is less than 1.

Step 1: Identify Coefficients

First, identify the coefficients of the given quadratic function \(f(x) = ax^2 + 2x + 1\). Here, \(b = 2\) and \(c = 1\) and \(a\) is the constant that we are trying to find

Step 2: Formulate Discriminant

Next, write down the discriminant formula which is found in the quadratic formula as \(b^{2} - 4ac\). Substitute \(b = 2\) and \(c = 1\) into the discriminant formula to get \(4 - 4a\)

Step 3: Solve for a

Now, as we want the function to have two distinct real roots, the discriminant should be greater than zero. So, set \(4 - 4a > 0\) and solve for \(a\). This should give \(a < 1\)

Step 4: Conclusion

So, for any value of \(a\) that is less than 1, the function will have two real zeros.

Key Concepts

Real Zeros

Real zeros, often referred to as real roots or solutions of a quadratic function, are the points where the graph of the function intersects the x-axis. This means the values of the variable for which the function equals zero. For a quadratic function, which is typically in the form of \(ax^2 + bx + c = 0\), there can be zero, one, or two real zeros. If there are two real zeros, they are the x-values where the graph crosses or touches the x-axis twice. If the graph touches the x-axis at only one point, we say the quadratic has one real zero, often called a repeated or double root. This concept is of great importance in understanding how the parabola behaves relative to the x-axis.

Discriminant

The discriminant is a key component in the quadratic formula. It is expressed as \(b^2 - 4ac\) and crucially determines the nature and number of roots of a quadratic equation.

• If the discriminant is greater than zero, the quadratic equation has two distinct real roots.
• If it is zero, the equation has exactly one real root, meaning the parabola touches the x-axis at one point.
• Finally, a negative discriminant means the quadratic equation has no real roots, but two complex ones.

In the exercise above, we set the discriminant greater than zero to ensure there are two distinct real zeros. This approach helps us find under what conditions, specifically the value of \(a\), the function will intersect the x-axis at two points.

Quadratic Equation

A quadratic equation is any equation that can be rearranged into the standard form \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are known as the coefficients. The solutions to a quadratic equation are found using various methods, such as factoring, using the quadratic formula, or completing the square. However, the quadratic formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\) is most commonly used, especially when an equation cannot be factored easily. The presence of \( \pm \sqrt{b^2-4ac} \) in this formula relates directly to the characteristics derived from the discriminant.

Coefficients

Coefficients are the numerical or constant factors that multiply the variables in an equation. In our quadratic equation \(f(x) = ax^2 + bx + c\), the coefficients are represented by \(a\), \(b\), and \(c\). In this particular function:

• \(a\) is the coefficient of \(x^2\), influencing the direction and steepness of the parabola.
• \(b\) is the coefficient of \(x\), which affects the parabola's axis of symmetry.
• \(c\) is the constant term, which provides the y-intercept of the parabola.

Understanding coefficients is crucial for interpreting the graph's geometry and analyzing the relationships between variables in quadratic functions.