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Chapter 42: Problem 1208

Find two real numbers whose sum is 10 such that the sum of the larger and the square of the smaller is 40 .

Short Answer

Expert verified
The two possible solutions for the real numbers are (-5, 15) and (6, 4).

Step by step solution

01

Define Variables and Write Equations

Let x be the smaller number and y be the larger number. We are given that their sum is 10, so we can write the first equation as: \(x + y = 10\) We are also given that the sum of the larger number and the square of the smaller number is 40, so we can write the second equation as: \(y + x^2 = 40\)
02

Solve the System of Equations

Now that we have two equations involving x and y, we can solve them using substitution or elimination method. In this case, it's easiest to use the substitution method. Solve the first equation for x in terms of y: \(x = 10 - y\) Now substitute this expression for x into the second equation: \(y + (10 - y)^2 = 40\)
03

Solve the Quadratic Equation

Expand and simplify the equation: \(y + (10 - y)^2 = y + (100 - 20y + y^2) = 40\) Combine the terms: \(y^2 - 19y + 100 = 40\) Subtract 40 from both sides to set the equation equal to zero: \(y^2 - 19y + 60 = 0\) Now use the quadratic formula or factoring method to find the possible values of y: \((y - 15)(y - 4) = 0\) Thus, y can be either 15 or 4.
04

Find the Corresponding x Values

For each value of y, find the corresponding x value by substituting y into the expression x = 10 - y: 1. If y = 15, x = 10 - 15 = -5 2. If y = 4, x = 10 - 4 = 6
05

Check the Solutions

Check if both obtained pairs of numbers (x, y) satisfy the given conditions: 1. For x = -5 and y = 15, \(x + y = -5 + 15 = 10\) and \(y + x^2 = 15 + (-5)^2 = 15 + 25 = 40\), so this pair of numbers satisfies the given conditions. 2. For x = 6 and y = 4, \(x + y = 6 + 4 = 10\) and \(y + x^2 = 4 + 6^2 = 4 + 36 = 40\), so this pair of numbers also satisfies the given conditions. Since both pairs satisfy the given conditions, there are two possible solutions: (x, y) = (-5, 15) and (x, y) = (6, 4).

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