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Chapter 2: Problem 99

Show that there do not exist two real numbers whose sum is 7 and whose product is 13 .

Short Answer

Expert verified
We assumed that there exist two real numbers \(x\) and \(y\) such that their sum is 7 and their product is 13. This led to a quadratic equation: \(x^2 - 7x + 13 = 0\). However, the discriminant of this equation, \(D = -3\), is negative, which means this equation has no real roots. Thus, we have reached a contradiction, and we conclude that there do not exist two real numbers whose sum is 7 and whose product is 13.

Step by step solution

01

Define variables for the numbers involved

Let x and y be the two real numbers we are trying to find. Our given statements about these numbers are: 1. x + y = 7 2. xy = 13
02

Solve one of the equations for one of the variables

We will solve the first equation for y: y = 7 - x
03

Substitute the expression for y in the second equation

Now, substitute the expression we found for y into the second equation (the product equation): x(7 - x) = 13
04

Simplify the equation and solve for x

Expand the equation and solve for x: 7x - x^2 = 13 Move all terms to the left of the equation: x^2 - 7x + 13 = 0
05

Check if the quadratic equation has real roots

To check if the quadratic equation has real roots, we need to calculate the discriminant: Discriminant (D) = b^2 - 4ac In our case, a = 1, b = -7, and c = 13. Plug these values into the discriminant formula: D = (-7)^2 - 4(1)(13) D = 49 - 52 D = -3
06

Analyze the result and draw the conclusion

Since the discriminant D is negative, the given quadratic equation does not have real roots. That means there is no real value of x that satisfies the equation, and consequently, there is no real value of y that satisfies the equation either. Thus, we have shown that there do not exist two real numbers whose sum is 7 and whose product is 13.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Discriminant
The discriminant in a quadratic equation helps us determine the nature of the roots without actually solving the equation. In a standard quadratic equation of the form \( ax^2 + bx + c = 0 \), the discriminant \( D \) is calculated using the formula:
  • \( D = b^2 - 4ac \)
The value of the discriminant gives us key insights:
  • If \( D > 0 \), the equation has two distinct real roots.
  • If \( D = 0 \), the equation has exactly one real root, also known as a repeated or double root.
  • If \( D < 0 \), the equation does not have real roots; instead, it has two complex roots.
In this exercise, the calculated discriminant was \( -3 \). Since it is negative, it tells us that the quadratic equation has no real roots. This supports the conclusion that no real numbers exist with the given sum and product.
Real Numbers
Real numbers include all the numbers that exist on the continuous number line, covering both rational and irrational numbers.
  • Rational numbers can be expressed as fractions or ratios, such as \( \frac{1}{2} \) or \( 3 \).
  • Irrational numbers cannot be neatly expressed as fractions, such as \( \sqrt{2} \) or \( \pi \).
Real numbers are crucial in solving equations since we often restrict our solutions to this set unless otherwise noted. In the exercise, we sought real numbers whose sum is 7 and product is 13. However, the negative discriminant indicated that our solutions, if they exist, are not real. Therefore, they lie outside the number line in the realm of complex numbers, which include an imaginary component.
Sum and Product of Numbers
When given two numbers, their sum and product can yield significant information, often used in quadratic equations. Suppose we know two numbers: \( x \) and \( y \).
  • Their sum is expressed as \( x + y \).
  • Their product is expressed as \( xy \).
These relationships were critical in setting up this exercise:
  • We set \( x + y = 7 \), expressing their combined value.
  • We set \( xy = 13 \), identifying how they multiply together.
These equations help form a quadratic equation when trying to solve for the individual values of \( x \) and \( y \). Rearranging these equations, and through substitution, we derived a quadratic form. Unfortunately, the discriminant analysis showed that such numbers, fitting both criteria within the real number system, do not exist. This peculiar result highlights that not all numerical combinations have solutions rooted in real numbers.

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Most popular questions from this chapter

Explain why $$ \sqrt{x}<\sqrt[3]{x} \quad \text { if } \quad 0\sqrt[3]{x} \quad \text { if } \quad x>1 $$ Sketch the graphs of the functions \(\sqrt{x}\) and \(\sqrt[3]{x}\) on the interval [0,4] .

Sketch the graph of the given function \(f\) on the domain \(\left[-3,-\frac{1}{3}\right] \cup\left[\frac{1}{3}, 3\right]\) $$ f(x)=-\frac{2}{x^{2}}+3 $$

Explain why the equation \(\sqrt{x^{2}}=x\) is not valid for all real numbers \(x\) and should be replaced by the equation \(\sqrt{x^{2}}=|x|\)

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Find a formula for the inverse function \(f^{-1}\) of the indicated function \(f\). $$ f(x)=x^{9} $$
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