Problem 28 - Elementary Algebra

Question

For the following problems, write the values of \(a, b,\) and \(c\) in quadratic equations. $$ x^{2}=9 $$

Step-by-Step Solution

Answer

Question: Identify the values of \(a\), \(b\), and \(c\) in the given quadratic equation \(x^2 - 9 = 0\). Answer: \(a = 1\), \(b = 0\), \(c = -9\)

Step 1: Rewrite the given equation as a standard quadratic equation

We can rewrite the given equation to resemble a standard quadratic equation by subtracting 9 from both sides. This will give us the equation in the form \(ax^2 + bx + c = 0\). So, \(x^2 - 9 = 0\)

Step 2: Identify the values of \(a\), \(b\), and \(c\)

Now, compare the rewritten equation \(x^2 - 9 = 0\) with the standard form \(ax^2 + bx + c = 0\). You can see that the coefficient of \(x^2\) is \(1\), so \(a = 1\). There is no \(x\) term in our equation, which means the coefficient of \(x\) (or \(b\)) is \(0\), so \(b = 0\). Finally, the constant term in our equation is \(-9\), so \(c = -9\). Therefore, the values of \(a\), \(b\), and \(c\) in the given quadratic equation are \(a = 1\), \(b = 0\), and \(c = -9\).

Key Concepts

Standard Form

To fully understand quadratic equations, let's start with the concept of the standard form. A quadratic equation is typically expressed in what we call "standard form." This format is structured as follows:

• \(ax^2 + bx + c = 0\)

Where:

• \(a\), \(b\), and \(c\) are constants, with \(a eq 0\).
• \(x\) represents the variable or unknown that we solve for.

For an equation to be in standard form, all terms should be on one side, equaling zero. This arrangement makes it easier to identify the key components for solving the equation. Let's use it in the equation \(x^2 = 9\) by rearranging it to \(x^2 - 9 = 0\). This setup allows for easier manipulation and solution finding.
To summarize, rewriting in standard form highlights the equation's structure and prepares it for analysis. Once you practice transforming equations, recognizing and working with them will become second nature.

Coefficients

Coefficients are vital parts of quadratic equations because they determine the equation's characteristics and solutions. In the standard form \(ax^2 + bx + c = 0\), the coefficients are:

• \(a\): the coefficient of \(x^2\), which controls the direction and width of the parabola.
• \(b\): the coefficient of \(x\), impacting the position of the parabola along the x-axis.
• \(c\): the constant term that affects the vertical shift of the parabola.

These coefficients shape the graph of the quadratic equation, influencing whether it opens up or down and how steep or wide it is. Knowing the coefficients \(a = 1\), \(b = 0\), and \(c = -9\) in \(x^2 - 9 = 0\) will help determine its characteristics.
Without coefficients, it becomes nearly impossible to predict the behavior of a quadratic equation. Understanding how they interact naturally leads to grasping the broader concepts of quadratic curves and their roles in various equations.

Quadratic Formula

The quadratic formula is a key tool for solving any quadratic equation written in standard form. It helps us find the roots, or solutions, of the equation. The formula is:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

This formula provides us two potential solutions: one with the plus sign and one with the minus sign. These roots represent the points where the graph of the quadratic equation crosses the x-axis.
The expression \(b^2 - 4ac\) is called the discriminant. It determines the nature of the equation's roots:

• If \(b^2 - 4ac > 0\), there are two distinct real roots.
• If \(b^2 - 4ac = 0\), there is exactly one real root.
• If \(b^2 - 4ac < 0\), there are no real roots (the roots are complex numbers).

For the equation \(x^2 - 9 = 0\), applying the quadratic formula is quick as \(b = 0\) simplifies the process. Understanding and applying this formula unlocks the solutions to diverse quadratic equations, even those that resist simpler approaches like factoring.