Problem 15 - Calculus for Biology and Medicine

Question

Explain why \(y=x^{2}-4\) has at least two roots.

Step-by-Step Solution

Answer

The function has two roots: \(x = -2\) and \(x = 2\).

Step 1: Understand the Definition of Roots

Roots of a function are the values of \(x\) for which the function equals zero. For the function \(y = x^2 - 4\), we are looking for \(x\) values that make \(y = 0\).

Step 2: Set the Function to Zero

We set the function equation to zero to find the roots. This gives us the equation \(x^2 - 4 = 0\).

Step 3: Simplify the Equation

Notice that \(x^2 - 4\) is a difference of squares. We can factor it as \((x + 2)(x - 2) = 0\).

Step 4: Solve for x

Setting each factor equal to zero, we have \(x + 2 = 0\) or \(x - 2 = 0\). Solving these gives us \(x = -2\) and \(x = 2\).

Step 5: Verify the Number of Roots

By solving the factors, we found two distinct solutions: \(x = -2\) and \(x = 2\). This confirms that the function has at least two roots.

Key Concepts

Roots of a Function

When dealing with quadratic equations, understanding the concept of the "roots of a function" is crucial. Simply put, the roots are the solutions to the equation when the quadratic function equals zero. When seeking the roots for a quadratic function like \(y = x^2 - 4\), we aim to find the \(x\) values that make the equation true when the result is zero.

To identify these roots, set the equation \(x^2 - 4\) equal to zero. Through this process, you investigate the points where the function intersects or touches the x-axis on a graph. Every intersection point represents a root of the function.

In essence, finding the roots means solving for \(x\) in \(x^2 - 4 = 0\), revealing that the function touches the x-axis at these points, illustrating the solutions where \(y\) or the output value is zero.

Difference of Squares

The term "difference of squares" describes a specific mathematical pattern where a square number is subtracted from another square number. An equation reflecting a difference of squares follows the form \(a^2 - b^2\) and can be quickly recognized by its easily factorable structure.

In the expression \(x^2 - 4\), we see that it's a difference of squares: \(x^2\) being the square of \(x\) and \(4\) being \(2^2\). Recognizing this pattern enables us to factor the equation efficiently into \((x + 2)(x - 2)\).

This pattern is useful because it turns a complex quadratic equation into a simpler multiplication, making solving possible roots much more straightforward. By identifying \((x + 2)(x - 2)\) = 0, we leverage the property of zero products, facilitating the solution for \(x = -2\) and \(x = 2\).

Factoring Quadratics

"Factoring quadratics" is a technique employed to simplify quadratic expressions, making it easier to identify the roots. The goal is to express a quadratic equation in a product form.

Take \(x^2 - 4\) for instance. Recognizing that this equation is a difference of squares, it can be factored into \((x + 2)(x - 2)\). Factoring transforms the equation into a multiplication of simpler binomials.

Once factored, the equation \((x + 2)(x - 2) = 0\) allows us to use the zero-product property, where each factor is set to zero:

• \(x + 2 = 0\), yielding \(x = -2\).
• \(x - 2 = 0\), resulting in \(x = 2\).

Factoring quadratics is a powerful technique not only to find the roots but to gain a deeper understanding of the structure and properties of quadratic functions.