Problem 65 - Elementary Algebra Within Reach

Question

In Exercises \(63-66\), solve the equation. $$ z^{2}-4 z-12=0 $$

Step-by-Step Solution

Answer

The solutions to the equation \(z^{2}-4z-12=0\) are \(z = 6\) and \(z = -2\).

Step 1: Identify a, b, and c

From the given quadratic equation \(z^{2} - 4z -12 = 0\), identify \(a = 1\), \(b = -4\), and \(c = -12\).

Step 2: Calculate Discrepant (\(b^2-4ac\))

Before we can use the quadratic formula, we need to calculate the value of the discrepant: \(b^{2} - 4ac = (-4)^{2} - 4(1)(-12) = 16 + 48 = 64\).

Step 3: Substitute a, b, and c into the Quadratic Formula

Substitute \(a=1\), \(b=-4\), and \(c=-12\) into the quadratic formula to find the roots: \(z = \frac{-(-4) \pm \sqrt{64}}{2(1)} = \frac{4 \pm 8}{2}\).

Step 4: Solve for z

We get two solutions for z, \(z = \frac{4 + 8}{2} = 6\) and \(z = \frac{4 - 8}{2} = -2\) are the roots of the given quadratic equation.

Key Concepts

Quadratic Formula

The quadratic formula is an essential tool for solving quadratic equations of the form \( ax^2 + bx + c = 0 \). This formula allows us to find the roots of any quadratic equation by simply plugging in the values of \( a \), \( b \), and \( c \).

The formula itself is expressed as:

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

This formula provides solutions in terms of \( x \) as soon as you substitute the coefficients.
Remember that \( a \), \( b \), and \( c \) are the numerical coefficients in your quadratic equation. Once these are identified, you can direct your attention to calculating the discriminant to proceed further using the formula.

Discriminant

The discriminant is a critical part of the quadratic formula, as it determines the nature of the roots. Found under the square root in the quadratic formula, it is represented as:

• \( b^2 - 4ac \)

The value of the discriminant provides insight into the number and type of roots.

• If \( b^2 - 4ac = 0 \), there is exactly one real root.
• If \( b^2 - 4ac > 0 \), there are two distinct real roots.
• If \( b^2 - 4ac < 0 \), the equation has two complex roots.

In our task, the discriminant was \( 64 \), a positive number, indicating two distinct real roots. Understanding the discriminant helps you grasp what to expect from the solutions of the quadratic equation.

Roots of a Polynomial

In solving quadratic equations, finding the roots is essentially finding the values of \( z \) (or \( x \), etc.) that satisfy the equation \( ax^2 + bx + c = 0 \). These roots are also known as the solutions of the polynomial. Essentially, they are the values where the polynomial equals zero.

Using the quadratic formula, we derived two roots for the equation \( z^{2} - 4z - 12 = 0 \):

• \( z = 6 \)
• \( z = -2 \)

These roots indicate the points where the graph of the polynomial intersects the horizontal line (\( z \)-axis) when plotted.
The concept of roots is vital because it tells you about the behavior of the polynomial and its graphical representation. Comprehending this connection aids in understanding how changes to the polynomial's coefficients alter its shape and position on the graph.