Problem 22 - College Algebra with Modeling and Visualization

Question

Exercises \(1-28:\) Solve the quadratic equation. Check your answers for Exercises \(1-12\). $$ -4 z^{2}+z+1=0 $$

Step-by-Step Solution

Answer

The solutions are \( z = \frac{1 - \sqrt{17}}{8} \) and \( z = \frac{1 + \sqrt{17}}{8} \).

Step 1: Identify the standard form

The given quadratic equation is \[ -4z^2 + z + 1 = 0 \]To solve this equation, we first confirm that it matches the standard form of a quadratic equation, which is:\[ ax^2 + bx + c = 0 \]Here, \( a = -4 \), \( b = 1 \), and \( c = 1 \).

Step 2: Use the quadratic formula

The quadratic formula to find the roots of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:\[ z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Substitute the values of \( a \), \( b \), and \( c \):\[ z = \frac{-1 \pm \sqrt{1^2 - 4(-4)(1)}}{2(-4)} \]

Step 3: Calculate the discriminant

The discriminant is the part under the square root in the quadratic formula: \[ b^2 - 4ac \]Calculate it using the given values:\[ 1^2 - 4(-4)(1) = 1 + 16 = 17 \]Since the discriminant is positive, there will be two distinct real roots.

Step 4: Solve for the roots

Substitute the discriminant back into the quadratic formula:\[ z = \frac{-1 \pm \sqrt{17}}{-8} \]This gives us two potential solutions:\[ z_1 = \frac{-1 + \sqrt{17}}{-8} \]and \[ z_2 = \frac{-1 - \sqrt{17}}{-8} \]

Step 5: Simplify the solutions

Simplify each solution:1. For \( z_1 \): \[ z_1 = -\frac{-1 + \sqrt{17}}{8} = \frac{1 - \sqrt{17}}{8} \]2. For \( z_2 \): \[ z_2 = -\frac{-1 - \sqrt{17}}{8} = \frac{1 + \sqrt{17}}{8} \]

Step 6: Check the solutions

Substitute each value back into the original equation to verify they hold true. For \( z_1 = \frac{1 - \sqrt{17}}{8} \) and \( z_2 = \frac{1 + \sqrt{17}}{8} \), the calculations show they satisfy the original equation, confirming our solutions are correct.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for finding the roots of any quadratic equation. A quadratic equation typically follows a general form:

• \( ax^2 + bx + c = 0 \)

Here, \( a \), \( b \), and \( c \) are constants, and the variable \( x \) represents an unknown. The quadratic formula itself is given by:

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

This formula is particularly useful because it provides a straightforward way to calculate the roots (solutions) of any quadratic equation.
All you need to do is plug in your \( a \), \( b \), and \( c \) values, and it does the heavy lifting for you.
Note the \( \pm \) sign in the formula: it tells us there are potentially two solutions. These solutions can be real or complex numbers.

Discriminant Calculation

The discriminant of a quadratic equation is a crucial component that can tell us a lot about the equation's roots without even solving for them. It's the part of the quadratic formula under the square root:

• \( b^2 - 4ac \)

This small yet significant calculation can indicate:

• If positive, the equation has two distinct real roots.
• If zero, the equation has exactly one real root (a repeated root).
• If negative, the equation has two complex roots.

For example, with our given equation \( -4z^2 + z + 1 = 0 \), calculating the discriminant gives \( 17 \), which is positive. This tells us that there will be two distinct real roots to find.
This alone provides insight into the nature of the solutions even before calculating them.

Roots of a Quadratic Equation

Finding the roots of a quadratic equation involves determining the values of \( x \) that satisfy \( ax^2 + bx + c = 0 \). These values, known as the "roots" or "solutions," are where the quadratic function intersects the x-axis when graphed. Using the quadratic formula is one way to find these roots:

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

Applying this to our specific equation \(-4z^2 + z + 1 = 0\):

• First root: \( z_1 = \frac{-1 + \sqrt{17}}{-8} = \frac{1 - \sqrt{17}}{8} \)
• Second root: \( z_2 = \frac{-1 - \sqrt{17}}{-8} = \frac{1 + \sqrt{17}}{8} \)

These outcomes are the points where the parabola representing the equation crosses the x-axis.
The roots are not just arbitrary numbers; they reveal important features about the quadratic equation's graph and how the equation behaves.