Problem 55 - Algebra for College Students

Question

Expressing solutions to the nearest one-thousandth. $$x^{2}-16 x-24=0$$

Step-by-Step Solution

Answer

The solutions rounded to the nearest one-thousandth are 17.381 and -1.381.

Step 1: Identify the equation

We are given the quadratic equation \( x^2 - 16x - 24 = 0 \). Our task is to solve this equation and express the solutions to the nearest one-thousandth.

Step 2: Identify coefficients

For the quadratic equation \(ax^2 + bx + c = 0\), identify \(a = 1\), \(b = -16\), and \(c = -24\).

Step 3: Apply the quadratic formula

The quadratic formula is \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\). Substitute values to get \(x = \frac{{-(-16) \pm \sqrt{{(-16)^2 - 4 \cdot 1 \cdot (-24)}}}}{2 \cdot 1}\).

Step 4: Calculate the discriminant

First, calculate the discriminant \(b^2 - 4ac\). This is \((-16)^2 - 4 \cdot 1 \cdot (-24) = 256 + 96 = 352\).

Step 5: Calculate the square root of the discriminant

Find \(\sqrt{352}\), which is approximately 18.761663. You'll need this in the quadratic formula.

Step 6: Find the two solutions

Substitute the value of \(\sqrt{352}\) back into the quadratic formula to find \(x\):- Positive solution: \(x = \frac{16 + 18.761663}{2} = 17.3808315\).- Negative solution: \(x = \frac{16 - 18.761663}{2} = -1.3808315\).

Step 7: Round the solutions

Round each solution to the nearest one-thousandth: - Positive root: 17.381 - Negative root: -1.381.

Key Concepts

Quadratic Formula

The quadratic formula is a reliable method for finding solutions to a quadratic equation of the form \( ax^2 + bx + c = 0 \). It provides a foolproof way to find the values of \( x \) that satisfy the equation. The formula is expressed as:\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\]with:

• "\(b\)" as the linear coefficient
• "\(a\)" as the coefficient of \(x^2\)
• "\(c\)" as the constant term

To use the quadratic formula effectively, follow these simple steps:
1. Identify the coefficients \(a\), \(b\), and \(c\) from the given quadratic equation.
2. Substitute these coefficients into the formula.
3. Calculate the value under the square root (the discriminant) and continue to find the results of the formula.
This method is powerful because it works for any quadratic equation, even when the factors aren't easily found. When you substitute into the formula, you will get two potential solutions for \(x\), represented as "\(+\)" and "\(-\)" in the equation. These solutions can represent the points at which the parabola intersects the x-axis.

Rounding Numbers

Rounding numbers, especially when dealing with decimal solutions, is a necessary skill in mathematics. When rounding to the nearest one-thousandth, we are focusing on the third digit after the decimal point. Here's how to do it effectively:

• Identify the digit in the one-thousandth place.
• Look at the digit immediately to the right.
• If the digit is 5 or greater, increase the one-thousandth digit by 1.
• If it’s less than 5, keep the one-thousandth digit the same.

For example, if you have 17.3808315, the one-thousandth position is 0. The digit to the right is 8, so you would round up to 17.381.
This process is vital whenever precision is needed in calculations, ensuring accuracy in the solutions derived from mathematical expressions.

Discriminants

The discriminant in a quadratic equation provides crucial insights into the nature of the roots of the equation. It is found using the expression \(b^2 - 4ac\) from the quadratic formula.

• A positive discriminant indicates that there are two distinct real roots.
• A zero discriminant means there is exactly one real root, known as a repeated or double root.
• A negative discriminant signifies there are two complex roots, meaning the roots are not real numbers.

In the given equation \(x^2 - 16x - 24 = 0\), the discriminant was calculated as 352, a positive number.
This indicates that there are two distinct real roots. Knowing about the discriminant enables us to predict the sort of solutions we can expect before actually solving the equation, which is especially helpful in plotting or understanding the behavior of the quadratic graphically.