Problem 62 - Intermediate Algebra

Question

Another of your friends claims that the quadratic formula can be used to solve the equation \(x^{2}-9=0\). How would you react to this claim?

Step-by-Step Solution

Answer

Yes, the quadratic formula can be used to solve \(x^2 - 9 = 0\). The solutions are \(x = 3\) and \(x = -3\).

Step 1: Introduction of the problem

We are presented with the equation \(x^2 - 9 = 0\). Our task is to determine if the quadratic formula can be used to solve this equation.

Step 2: Understanding the quadratic formula

The quadratic formula is used to find the roots of a quadratic equation of the form \(ax^2 + bx + c = 0\). It is given by the formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Step 3: Transform the equation to general quadratic form

Our equation, \(x^2 - 9 = 0\), can be compared to the general quadratic form \(ax^2 + bx + c = 0\). For this equation, we identify \(a = 1\), \(b = 0\), and \(c = -9\).

Step 4: Applying the quadratic formula

Substitute \(a = 1\), \(b = 0\), and \(c = -9\) into the quadratic formula:\[ x = \frac{-0 \pm \sqrt{0^2 - 4 \cdot 1 \cdot (-9)}}{2 \cdot 1} \]Simplifying the expression:\[ x = \frac{\pm \sqrt{36}}{2} \]\[ x = \frac{\pm 6}{2} \]\[ x = \pm 3 \]

Step 5: Consider alternative methods

Alternatively, we could solve \(x^2 - 9 = 0\) by factoring the left side as a difference of squares:\((x - 3)(x + 3) = 0\) This results in the solutions \(x = 3\) and \(x = -3\).

Key Concepts

Quadratic Equations

Quadratic equations are fundamental in algebra and have the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). These equations form a parabola when graphed in a coordinate plane, opening either upwards when \(a > 0\) or downwards when \(a < 0\).

Certain characteristics define quadratic equations, one of which is the degree, specifically a degree of two due to the highest power being \(x^2\).

• The coefficient \(a\) determines the width and direction of the parabola.
• \(b\) affects the symmetry and the location of the vertex along the x-axis.
• The constant \(c\) represents the y-intercept.

To solve quadratic equations, we often use methods such as factoring, completing the square, or applying the quadratic formula. Each method has its specific applications depending on the form in which the quadratic equation is presented. Understanding these principles will allow students to choose the most efficient method for solving particular problems.

Factoring

Factoring in quadratic equations involves expressing the equation as a product of its linear factors. This method is particularly useful when the quadratic can be easily decomposed into simple binomial expressions, as shown in the example with \(x^2 - 9 = 0\).

By recognizing this as a difference of squares, we rewrite it:
\((x - 3)(x + 3) = 0\)
This provides us with two solutions: \(x = 3\) and \(x = -3\).

The core principle behind factoring is the zero-product property, where if a product of two expressions equals zero, at least one of the expressions must equal zero.

• Factoring is often faster than other methods and is preferred when applicable.
• Identifying common factors can simplify the process prior to reaching the difference of squares form.
• Not all quadratics will be easily factorable, in which case one might rely on the quadratic formula or completing the square.

Roots of Equations

The roots of a quadratic equation refer to the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). These roots are the x-values at which the graph of the parabola intersects the x-axis.

Depending on the discriminant (\(b^2 - 4ac\)), the roots of a quadratic equation can be real or complex:

• If \(b^2 - 4ac > 0\), there are two distinct real roots.
• If \(b^2 - 4ac = 0\), there is one real root, also known as a repeated or double root.
• If \(b^2 - 4ac < 0\), the roots are two complex numbers and the graph does not cross the x-axis.

For the quadratic \(x^2 - 9 = 0\), the discriminant is \(0^2 - 4 \cdot 1 \cdot (-9) = 36\), a positive value indicating two distinct real roots: \(x = 3\) and \(x = -3\).

Acquiring a clear understanding of how to determine the roots enables students to solve quadratic equations by whichever method proves most effective in different scenarios.