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Find two integers whose a. product is 10 and whose sum is 7 . b. product is 8 and whose sum is $-6$ c. product is $-6$ and whose sum is 1 d. product is $-9$ and whose sum is $-8 .$

   Find two integers whose
a. product is 10 and whose sum is 7 .
b. product is 8 and whose sum is $-6$
c. product is $-6$ and whose sum is 1
d. product is $-9$ and whose sum is $-8 .$

Elementary and Intermediate Algebra

Elementary and Intermediate Algebra

Alan S. Tussy, R.… 5th Edition

Chapter 6, Problem 10

Instant Answer

Solved by Expert Jessica Harter

03/25/2020

Step 1

We can think of the numbers 2 and 5, because 2*5 = 10 and 2+5 = 7. So, the two numbers are 2 and 5. Show more…

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Find two integers whose a. product is 10 and whose sum is 7 . b. product is 8 and whose sum is $-6$ c. product is $-6$ and whose sum is 1 d. product is $-9$ and whose sum is $-8 .$

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Key Concepts

System of Equations

This concept involves setting up simultaneous equations based on the relationships provided by the sum and product of two numbers. By representing the two unknown integers with variables, you create a system that can be solved using various methods, allowing the translation of verbal conditions into mathematical equations.

Quadratic Equations

When given the sum and product of two numbers, a quadratic equation can be formulated by leveraging the relationships x + y = S and xy = P, leading to the equation x² - Sx + P = 0. Solving this quadratic equation allows you to discover the integers that satisfy both conditions.

Factor Pair Analysis

This method involves listing all possible pairs of integers that multiply to the given product and then checking which pair of integers also sums to the given value. This approach often simplifies the process of finding integer solutions without resorting to the quadratic formula, especially when the numbers involved are small.

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Transcript

00:01 So we have parts a, b, c, and d, and i already have them all written out here.

00:05 And so for each problem, we're trying to find two numbers that multiply to something and then add to another number.

00:14 So for part a here, we're finding two numbers that multiply to 10, but add or whose sum is positive 7.

00:22 So let's just start thinking right away.

00:24 What are some ways that i can get 10? right away i just think of 2 and 5.

00:32 So 2 times 5 equals 10, and 2 plus 5, that equals 7.

00:41 So right off the bat, we found two numbers, 2 and 5, that when i multiply i get 10, and when i add them, i get 7.

00:51 Part b, kind of the same idea.

00:54 Two numbers that multiply to 8 and add to negative 6.

00:59 So, again, i'm just thinking, i know two times four equals eight.

01:05 Now let's check.

01:07 Two plus four equals six.

01:11 Not quite.

01:12 We'll just put an x there.

01:15 It's close.

01:16 It's positive six, though, and we want negative six.

01:20 So let's just try to change the sign of the original two numbers we thought of.

01:27 So let's try negative 2 times negative 4...

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