The sum of the squares of four consecutive integers is 86 . Find the four integers.

Name: Problem 32, Chapter 16: Prealgebra and Introductory Algebra
Uploaded: 2020-06-25T21:57:43-08:00
Duration: 3 min 42 s
Channel: Trinity Steen
Description: The sum of the squares of four consecutive integers is 86 . Find the four integers.

step 1 So this is an integer problem, a consecutive integer problem. So your first setup is to get your

The sum of the squares of four consecutive integers is 86 ....

Key Concepts

Consecutive Integers

This concept refers to numbers that follow one another in order, typically with a common difference of 1. In mathematical problems, consecutive integers are often represented by expressions like n, n+1, n+2, etc., allowing one to establish relationships between them and solve for unknowns.

Sum of Squares

The sum of squares involves taking several integers, squaring each one, and then adding those squares together. This concept is useful in various areas of mathematics and problem-solving, including statistics and algebra, as it transforms linear relationships into quadratic forms.

Algebraic Equation Formulation

This key concept involves translating a word problem into an algebraic equation by representing unknown quantities with variables and expressing given conditions mathematically. It is crucial for systematically solving problems by creating a clear equation that encapsulates all relationships described.

Quadratic Equations

When the algebraic formulation of a problem leads to equations involving a variable raised to the second power, the result is a quadratic equation. Solving quadratic equations, through factoring, completing the square, or using the quadratic formula, is a fundamental skill in algebra that allows for determining possible values of the variable.

Transcript

00:01 So this is an integer problem, a consecutive integer problem.

00:03 So your first setup is to get your variables figured out.

00:06 So you got your x is your first number.

00:08 X plus 1 is the number right after.

00:10 So that's your second number, x plus 2 will be third and x was 3 will be your fourth.

00:15 But that they give us the information for number 32 that the sum of the squares of these is going to equal 86.

00:21 So that means my setup is going to be x squared for that first integer plus x plus x plus 1 in parentheses squared plus x plus 2 in parentheses squared plus x plus 3 in parentheses squared is equal to the 86 and that then from there we're supposed to go and solve and figure out what the four integers are so let's go ahead and clean that up a little bit you're going to have x squared is fine here then you foil out x plus 1 squared so x plus 1 times x plus 1 foiled out gets us x squared plus x squared plus 2x plus 1 then you've got x plus 2 times x plus 2 foiled is going to get us x squared plus 4 x plus 4 then we're going to have the x plus 3 foiled out so that's going to give us x squared plus 6x plus 9 and that is still equal to the 86 so we're going to definitely need to combine some like terms get the simplified and get this simplified and this cleaned up.

01:36 So we've got x squared and x squared and x squared and x squared.

01:39 So that's four x squared's happening.

01:42 And then we've got a 2x, a 4x, and a 6x.

01:46 That's going to get us 12x.

01:49 And then we got a 1, a 4, and a 9.

01:52 That all combines the 14 is equal to 86...

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