Problem 29 - Precalculus : Building Concepts and Connections

Question

Find a possible expression for a quadratic function \(f(x)\) having the given zeros. There can be more than one correct answer. $$x=-3 \text { and } x=0$$

Step-by-Step Solution

Answer

So, one possible quadratic function with the given zeros is \(f(x) = x^{2} + 3x\)

Step 1: Understanding the problem

In this problem, we are given two zeros of a quadratic function, \(x = -3\) and \(x = 0\). When these values are plugged into the function, the result should be \(0\). That is, \(f(-3) = f(0) = 0\). Now we apply these into the quadratic equation \(f(x) = ax^{2} + bx + c\)

Step 2: Incorporate Zeros

First, substitute \(x = -3\) into the quadratic equation \(f(x) = ax^{2} + bx + c = 0\). This gives us \(a(-3)^{2} + b(-3) + c = 0\), which simplifies to \(9a - 3b + c = 0\). Secondly, doing this for \(x = 0\), we get \(a(0)^{2} + b(0) + c = 0\), which simplifies to \(c = 0\).

Step 3: Answer

Now we know that \(c = 0\), the equation simplifies to \(9a - 3b = 0\) by substituting the first equation. The terms \(a\) and \(b\) constitute the co-efficients of the function. We also know that \(a\) is not equal to zero for this to be a quadratic equation. Therefore, any equations where \(a ≠ 0\) and \(9a = 3b\) will suffice. We can take one such example, \(a = 1, b = 3\) and the quadratic equation will become \(f(x) = x^{2} + 3x\).

Key Concepts

Quadratic Equation

A quadratic equation is a second-degree polynomial equation in a single variable x, with a non-zero coefficient for the term with x squared. It generally takes the form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \).

When working with quadratic equations, it's important to understand its graph, which is a parabola. The points where this parabola crosses the x-axis are known as the 'roots' or 'zeros' of the equation. These values are what makes the entire equation equal to zero. Discovering these values, as we did by using the given zeros \( x = -3 \) and \( x = 0 \), is vital as it determines the shape and placement of the parabola on the graph.

So why is \( a eq 0 \) crucial? If \( a \) were zero, then the equation would not be quadratic, but rather a linear one, changing the entire nature of the problem. Hence, while the values of \( b \) and \( c \) can be zero, \( a \) must always be a non-zero value to preserve the 'quadratic' nature of the equation.

Factoring Quadratics

The process of factoring quadratics is a method used to express the quadratic equation in a product of two binomials. In simple terms, it is the reverse of expanding multiplication of binomials.

Factoring can be useful when attempting to solve quadratic equations, as it can reveal the roots of the equation. For example, if you have a quadratic function \( f(x) = x^2 + 3x \) as in our case, you might notice that there is a common factor, which is \( x \). Factoring out the \( x \) would give us \( f(x) = x(x + 3) \). The roots of this function can be clearly seen: when \( x = 0 \) or \( x = -3 \), the function equals zero.

To learn factoring effectively, identify common factors, and try pairing terms to make perfect squares or look for patterns such as the difference of squares. Factoring quadratics is essential algebraic skill that simplifies finding zeros and understanding the function's behavior.

Quadratic Roots

The quadratic roots, also referred to as zeros, are the values of \( x \) that make the quadratic equation equal to zero. These are the points where the graph of the quadratic equation intersects the x-axis.

The roots can be found by various methods, including factoring, using the quadratic formula, completing the square, or even graphing. In the given exercise, the roots \( x = -3 \) and \( x = 0 \) were provided to us directly, which helped in determining the quadratic function.

The concept of roots is crucial in understanding not only the nature of quadratic equations but also the intersection points in more practical applications like physics, economics, or anywhere else quadratic functions are used. Remember, a quadratic equation can have either two real roots, one real root (in the case of a perfect square), or two complex roots (when the discriminant is less than zero).