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Chapter 5: Problem 40

The sum of the squares of two negative numbers is 145 and the difference of the squares of the numbers is 17 . Find the numbers.

Short Answer

Expert verified
The numbers are -9 and -8.

Step by step solution

01

Setup the Equations

Let the two negative numbers be -a and -b. According to the problem, their squares add up to 145 and the difference of their squares is 17. This gives us two equations: 1) a^2 + b^2 = 145 2) a^2 - b^2 = 17
02

Use the Difference of Squares Formula

Use the difference of squares formula on equation 2: a^2 - b^2 = (a + b)(a - b) = 17
03

Add the Equations

Add the two equations together: a^2 + b^2 + a^2 - b^2 = 145 + 17 2a^2 = 162 Solving for a^2 gives a^2 = 81
04

Solve for a

Take the square root of both sides to find a. Since the numbers are negative, a = -9
05

Solve for b

Substitute a^2 back into one of the original equations to find b^2: 81 + b^2 = 145 b^2 = 64. Take the square root of both sides to find b. Since the numbers are negative, b = -8
06

Verify

Verify the solution by checking that the sum and difference of squares match the original conditions: ( -9 )^2 + ( -8 )^2 = 81 + 64 = 145 ( -9 )^2 - ( -8 )^2 = 81 - 64 = 17

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Key Concepts

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

Quadratic Equations
Quadratic equations are mathematical expressions where the highest exponent of a variable is 2. They typically take the form \( ax^2 + bx + c = 0 \).In this exercise, we have two quadratic equations derived from the conditions given: \( a^2 + b^2 = 145 \) and \( a^2 - b^2 = 17 \).These equations use squares of variables instead of linear terms. In general, quadratic equations can be solved using factoring, completing the square, or using the quadratic formula.Understanding how to manipulate and solve these types of equations is key to finding the unknowns in systems involving quadratic terms.
Difference of Squares
The difference of squares is a specific algebraic formula represented as \( a^2 - b^2 = (a+b)(a-b) \). It's incredibly useful for simplifying and solving quadratic equations. In this problem, recognizing this pattern allows us to transform \( a^2 - b^2 = 17 \) into a multiplicative form: \( (a + b)(a - b) = 17 \).
This transformation is pivotal as it simplifies solving using steps like addition, subtraction, or substitution.The ability to spot the difference of squares quickly in algebra helps in efficiently solving more complex equations.
Algebraic Verification
Algebraic verification involves checking the solutions of equations to ensure they satisfy the original conditions. In this problem, after finding \(a = -9\) and \(b = -8\), we substitute these back into the original equations to verify our solutions. This step confirms our findings:
  • \( (-9)^2 + (-8)^2 = 81 + 64 = 145 \)
  • \( (-9)^2 - (-8)^2 = 81 - 64 = 17 \)
Ensuring the left side equals the right side of the original equations confirms the correctness. Verification prevents mistakes and builds confidence in problem-solving.
Negative Numbers
Negative numbers can be trickier to deal with, especially in squaring operations. When squared, a negative number becomes positive, e.g., \( (-9)^2 = 81 \). This is because multiplying two negative numbers yields a positive result. In our problem, recognizing the negative nature of \(a \) and \(b \) (\(-9 \)and \(-8 \) respectively) was crucial to finding correct roots:
  • \( a = -9 \)
  • \( b = -8 \)
Working with negative numbers also requires careful attention to signs when performing arithmetic operations, ensuring correct solutions in algebraic expressions involving both positive and negative terms.

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Most popular questions from this chapter

A plant nursery sells two sizes of oak trees to landscapers. Large trees cost the nursery \(\$ 120\) from the grower. Small trees cost the nursery \(\$ 80\). The profit for each large tree sold is \(\$ 35\) and the profit for each small tree sold is \(\$ 30 .\) The monthly demand is at most 400 oak trees. Furthermore, the nursery does not want to allocate more than \(\$ 43,200\) each month on inventory for oak trees. a. Determine the number of large oak trees and the number of small oak trees that the nursery should have in its inventory each month to maximize profit. (Assume that all trees in inventory are sold.) b. What is the maximum profit? c. If the profit on large trees were \(\$ 50\), and the profit on small trees remained the same, then how many of each should the nursery have to maximize profit?

Josh makes \(\$ 24 /\) hr tutoring chemistry and \(\$ 20 / \mathrm{hr}\) tutoring math. Let \(x\) represent the number of hours per week he spends tutoring chemistry. Let \(y\) represent the number of hours per week he spends tutoring math. a. Write an objective function representing his weekly income for tutoring \(x\) hours of chemistry and \(y\) hours of math. b. The time that Josh devotes to tutoring is limited by the following constraints. Write a system of inequalities representing the constraints. \- The number of hours spent tutoring each subject cannot be negative. \- Due to the academic demands of his own classes he tutors at most \(18 \mathrm{hr}\) per week. \- The tutoring center requires that he tutors math at least 4 hr per week. \- The demand for math tutors is greater than the demand for chemistry tutors. Therefore, the number of hours he spends tutoring math must be at least twice the number of hours he spends tutoring chemistry. c. Graph the system of inequalities represented by the constraints. d. Find the vertices of the feasible region. e. Test the objective function at each vertex. f. How many hours tutoring math and how many hours tutoring chemistry should Josh work to maximize his income? g. What is the maximum income? h. Explain why Josh's maximum income is found at a point on the line \(x+y=18\).

Determine if the ordered pair is a solution to the system of equations. (See Example 1\()\) \(3 x-5 y=-7\) \(x-4 y=-7\) a. (1,2) b. \(\left(-\frac{2}{3}, 1\right)\)

Jonas performed an experiment for his science fair project. He learned that rinsing lettuce in vinegar kills more bacteria than rinsing with water or with a popular commercial product. As a follow-up to his project, he wants to determine the percentage of bacteria killed by rinsing with a diluted solution of vinegar. a. How much water and how much vinegar should be mixed to produce 10 cups of a mixture that is \(40 \%\) vinegar? b. How much pure vinegar and how much \(40 \%\) vinegar solution should be mixed to produce 10 cups of a mixture that is \(60 \%\) vinegar?

Two angles are supplementary. The measure of one angle is \(12^{\circ}\) more than 5 times the measure of the other angle. Find the measure of each angle.
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