The sum of two numbers is 20, and the sum of their squares is 218 . Find the numbers.

Name: Problem 58, Chapter 6: Beginning and Intermediate Algebra
Uploaded: 2020-06-18T16:14:43-08:00
Duration: 5 min 25 s
Channel: Meredith Moody
Description: The sum of two numbers is 20, and the sum of their squares is 218 . Find the numbers.

The sum of two numbers is 20, and the sum of their squares...

Key Concepts

Systems of Equations

Systems of equations involve multiple equations that must be satisfied by the same set of unknowns. In this problem, the relationship given by the sum of the numbers and the sum of their squares forms a system that can be solved to find the individual values.

Substitution Method

The substitution method is a strategy for solving systems of equations where one equation is solved for one variable and then substituted into the other equation. This simplifies the system to an equation in one variable, which can then be solved more easily.

Quadratic Equations

A quadratic equation is a second-degree polynomial equation commonly encountered in algebra. After substitution, the problem leads to a quadratic equation, the solution of which can yield the individual values of the unknown numbers.

Factoring

Factoring is a method used to solve quadratic equations by expressing the quadratic polynomial as a product of two linear factors. When applicable, this technique provides a straightforward way to determine the roots of the equation.

Transcript

00:01 In order to figure out what my two numbers are here in this little puzzle, i'm going to have to use some substitutions and write one variable in terms of the other so that i can create a quadratic that i can factor to find the answer.

00:18 So the first thing i'm going to do is i'm going to write a in terms of b.

00:23 So what that means is i'm going to say, okay, well, a equals and subtract b from both sides, 20 minus.

00:32 B.

00:33 So this is what i'm writing a in terms of b.

00:37 So instead of using the variable a to describe a, i'm now using the variable b to describe a.

00:43 And what that lets me do is that lets me substitute that into the equation where i have my squared, like some of the squares, and then it takes it from a two variable equation down to a one variable equation.

00:57 So this is what i mean by that.

00:59 So i'm going to substitute my a in terms of b as 20 minus b squared.

01:06 And then i'm just going to leave my other b squared right there.

01:09 So i have now this is my a, but my a in terms of b plus b squared.

01:14 And this is something i'm going to foil out.

01:20 20 minus b times 20 minus b plus b squared equals 218.

01:27 And so 20 times 20 is 400.

01:30 20 times negative b is minus 20b 20 b 20 minus or negative b times 20 is minus 20 b and negative b times negative b is positive b squared so plus b squared equals 218 so now i'm going to combine my leg terms as my b and then i have my b squared terms so i have 4009 minus 40 b so minus 20 minus 20 gives me minus 40 and then plus 2b squared oh let me fix that that does not turn out well here we go 2b squared equals 218 now i have to have my constant on the same side as everything else so i'm going to subtract 218 from both sides and that will leave me with 182 minus 40b plus 2b squared equals 0.

02:41 And then i'm going to rewrite this in my standard form where i have my a x squared plus bx plus c equals 0.

02:52 Now i can factor this.

02:55 Well, actually i can do something first.

02:56 So i have 2b squared minus 40b plus 182.

03:02 And what you might notice is that these have a common factor of two.

03:08 So i have two.

03:09 I'm going to divide that out and i'm going to be left with b squared here, 20b here, and 91 here...