Problem 42 - Algebra 2

Question

NUMBER THEORY Use a quadratic equation to find two real numbers that satisfy each situation, or show that no such numbers exist. Their sum is \(-9\) and their product is 24

Step-by-Step Solution

Answer

No real numbers satisfy both conditions.

Step 1: Understand the Problem

We need to find two numbers whose sum is -9 and whose product is 24. The problem can be represented as a system of equations: \( x + y = -9 \) and \( xy = 24 \).

Step 2: Express One Variable in Terms of the Other

Using the sum equation \( x + y = -9 \), we can express \( y \) in terms of \( x \): \( y = -9 - x \).

Step 3: Substitute into the Product Equation

Substitute \( y = -9 - x \) into the product equation \( xy = 24 \). The equation becomes \( x(-9 - x) = 24 \).

Step 4: Simplify the Equation

The equation \( x(-9 - x) = 24 \) simplifies to \( -x^2 - 9x = 24 \). Rearrange it to standard quadratic form: \( x^2 + 9x + 24 = 0 \).

Step 5: Solve the Quadratic Equation

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 9 \), and \( c = 24 \). This becomes \( x = \frac{-9 \pm \sqrt{9^2 - 4 \cdot 1 \cdot 24}}{2 \cdot 1} \).

Step 6: Calculate the Discriminant

Compute the discriminant \( b^2 - 4ac = 81 - 96 = -15 \).

Step 7: Determine the Nature of the Roots

Since the discriminant is negative (\(-15\)), the quadratic equation has no real solutions.

Key Concepts

Number Theory

Number theory, in its simplest form, deals with the properties and relationships between numbers. It is a branch of pure mathematics devoted to the study of integers and integer-valued functions.
In this exercise, we explored a problem involving two numbers whose sum and product needed to be determined with the help of a quadratic equation. The equations derived from these properties are:

• The sum equation: \(x + y = -9\)
• The product equation: \(xy = 24\)

The goal was to find two numbers (let's denote them as \(x\) and \(y\)) satisfying these conditions.

This approach uses algebraic techniques to relate and work with numbers, further showcasing number theory's utility in solving integer-related problems. Understanding how to set up and manipulate these equations is a key aspect of number theory.

Real Numbers

Real numbers include all the numbers on the number line, covering both rational and irrational numbers. In this exercise, the focus was on finding real numbers whose specific sum and product are known.

In the context of quadratic equations, the presence of a negative discriminant, as discovered in the solution, shows that no real number solutions exist for the equation. The discriminant, represented by \(b^2 - 4ac\), determines the nature of the roots of a quadratic equation:

• A positive discriminant indicates two distinct real roots.
• A discriminant of zero indicates a repeated real root.
• A negative discriminant, however, signifies no real roots.

In our case, a negative discriminant of \(-15\) reveals that the roots of our quadratic equation are not real, which aligns with the foundational properties of quadratic equations within the realm of real numbers.

Sum and Product of Roots

The sum and product of roots is an interesting property of quadratic equations that relates directly to the coefficients of the equation. Given the quadratic equation in the standard form \(ax^2 + bx + c = 0\), these properties state:

• The sum of the roots, \(x_1 + x_2\), is equal to \(-\frac{b}{a}\).
• The product of the roots, \(x_1 x_2\), is equal to \( \frac{c}{a}\).

In this problem, we had specific values: a sum \( x + y = -9\) and a product \( xy = 24\). These values can typically be used to construct a corresponding quadratic equation. However, upon solving the equation, we found that the sum and product did not correspond to real roots because the calculated discriminant was negative.

Understanding these properties not only aids in solving quadratic equations but also helps in recognizing when it is impossible to find real number solutions, as was demonstrated with the derivation and solving process in this exercise.