Problem 52 - Algebra for College Students

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

The quadratic formula can solve the equation, confirming your friend's claim.

Step 1: Identify the Equation

The given equation is \(x^2 - 9 = 0\). This is a quadratic equation because it is in the form \(ax^2 + bx + c = 0\). Here, \(a = 1\), \(b = 0\), and \(c = -9\).

Step 2: Analyze the Quadratic Formula

The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). For any quadratic equation in the standard form, this formula can be used to find the roots of the equation.

Step 3: Apply the Quadratic Formula

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

Step 4: Simplify the Expression

The solution can be simplified as follows: \(x = \frac{6}{2} = 3\) and \(x = \frac{-6}{2} = -3\). Thus, the roots of the equation are \(x = 3\) and \(x = -3\).

Step 5: Conclusion on the Claim

The claim that the quadratic formula can be used to solve \(x^2 - 9 = 0\) is correct. The roots \(x = 3\) and \(x = -3\) obtained using the quadratic formula show it is applicable. However, since this is a simple difference of squares, a simpler method could also solve it efficiently.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. Any equation in the form \( ax^2 + bx + c = 0 \) can be tackled by using this formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). It's derived from completing the square and always results in expressions that compute the roots of any quadratic equation.

One of the advantages of the quadratic formula is its universality. Whether the root is real or complex, or if there's a double root, you can still use it effectively. Here's a basic rundown of how it works:

• First, identify \( a \), \( b \), and \( c \) from your quadratic equation.
• Second, substitute these values into the formula.
• Evaluate the discriminant (\( b^2 - 4ac \)) to determine the number and type of roots.
• Solve the equation by completing the operations for \( x \).

While it is a remarkably versatile formula, certain equations like the difference of squares can be solved more simply using alternate methods. However, the quadratic formula ensures accuracy in every scenario.

Difference of Squares

The difference of squares is a fundamental concept in algebra that allows for simple factorizations. It looks like this: \( x^2 - a^2 \), which can be expressed as the product \( (x + a)(x - a) \). This fact can be used to solve equations without relying on more complex methods like the quadratic formula.

Recognizing a difference of squares involves:

• Noting the presence of a subtraction operation between two perfect squares.
• Knowing that the squares have no other terms besides \( x^2 \) and a constant that is a perfect square.

For the equation \( x^2 - 9 = 0 \), you identify it as a difference of squares because 9 is a perfect square (\( 3^2 \)). Thus, it can be factored as \( (x + 3)(x - 3) = 0 \). From this factorization, you immediately see that the solutions, or roots, are \( x = 3 \) and \( x = -3 \).
Using the difference of squares drastically simplifies the process by cutting out additional algebraic steps.

Roots of Equations

Finding the roots of an equation means identifying where the equation equals zero. For quadratic equations, this typically means solving for \( x \) where \( ax^2 + bx + c = 0 \). The solutions, known as roots, might be real or complex numbers depending on the equation involved.

There are various strategies for finding these roots, including:

• Using the quadratic formula, which operates broadly across quadratic equations.
• Factoring, when applicable, to identify roots quickly, particular in straightforward equations like the difference of squares.

The nature of the roots is highly dependent on the discriminant from the quadratic formula, which is \( b^2 - 4ac \).
- If the discriminant is positive, expect two distinct real roots.
- If zero, two coinciding real roots, known as a "double root."
- And if negative, expect two complex roots.
Effectively identifying the roots of equations not only solves them but also provides insight into the behavior of the corresponding quadratic function.