Problem 80 - Algebra for College Students

Question

Find the exact solution \((s)\) to each problem. If the solution(s) are irrational, then also find approximate solution(s) to the nearest tenth. More missing numbers. Find two real numbers that have a sum of 8 and a product of 2

Step-by-Step Solution

Answer

The numbers are \( 4 + \sqrt{14} \approx 7.7 \) and \( 4 - \sqrt{14} \approx 0.3 \).

Step 1: Define the Variables

Let the two numbers be represented by variables: let them be denoted as x and y.

Step 2: Set Up the Equations

Using the given conditions, create two equations: \( x + y = 8 \) (sum) and \( xy = 2 \) (product).

Step 3: Express One Variable in Terms of the Other

Solve the sum equation for y : \( y = 8 - x \).

Step 4: Substitute into the Product Equation

Replace y in the product equation with 8 - x: \( x(8 - x) = 2 \).

Step 5: Form a Quadratic Equation

Simplify the equation to get a standard quadratic form: \( x^2 - 8x + 2 = 0 \).

Step 6: Solve the Quadratic Equation Using the Quadratic Formula

The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, a = 1, b = -8, c = 2.\[ x = \frac{8 \pm \sqrt{64 - 8}}{2} \]\[ x = \frac{8 \pm \sqrt{56}}{2} \]\[ x = \frac{8 \pm 2\sqrt{14}}{2} \]\[ x = 4 \pm \sqrt{14} \]

Step 7: Approximate the Irrational Solutions

Find the decimal values: \( x = 4 + \sqrt{14} \approx 7.7 \) and \( x = 4 - \sqrt{14} \approx 0.3 \).

Key Concepts

System of Equations

A system of equations is a collection of two or more equations with the same set of variables.
In this exercise, we are given two conditions forming our system:

• The sum of two numbers is 8: \( x + y = 8 \)
• The product of the same two numbers is 2: \( xy = 2 \)

By solving this system, we can find the values of the unknown variables. Let’s break down the solution process:
1. **Express one variable in terms of the other**: From the sum equation, we can represent one variable in terms of the other, i.e., \( y = 8 - x \).
2. **Substitute into the other equation**: Replace \( y \) in the product equation with \( 8 - x \), leading to \( x(8 - x) = 2 \).
This substitution reduces the system of equations to a single quadratic equation that we can solve using methods like factoring, completing the square, or the quadratic formula.

Sum and Product of Roots

In quadratic equations of the form \( ax^2 + bx + c = 0 \), the sum and product of roots have special relationships with the coefficients.
For the quadratic equation obtained in the exercise, \( x^2 - 8x + 2 = 0 \):

• The sum of the roots \( (x_1 + x_2) \) is equal to \( -b/a = 8 \), matching our original sum condition.
• The product of the roots \( (x_1 \times x_2) \) is equal to \( c/a = 2 \), matching our original product condition.

These relationships can be derived from Vieta’s formulas, which are useful when dealing with any quadratic equation. Knowing these can often help verify our solutions or find missing roots quickly.

Irrational Solutions

The solutions to quadratic equations can sometimes result in irrational numbers, which are numbers that cannot be expressed as exact fractions and have non-repeating, non-terminating decimal expansions.
In our exercise, the solutions to the quadratic equation \( x^2 - 8x + 2 = 0 \) are:\( x = 4 + \sqrt{14} \) and \( x = 4 - \sqrt{14} \)
These solutions are irrational because \( \sqrt{14} \) is an irrational number. To find approximate values, we can evaluate these expressions using a calculator, getting:

• \( x = 4 + \sqrt{14} \approx 7.7 \)
• \( x = 4 - \sqrt{14} \approx 0.3 \)

Irrational solutions occur frequently when the discriminant \( b^2 - 4ac \) in the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) is not a perfect square. Understanding how to work with irrational solutions is crucial for precise and accurate mathematics.