The sum of a number and its square is 182 . Find the number(s).

Name: Problem 22, Chapter 6: Beginning and Intermediate Algebra
Uploaded: 2020-06-09T15:47:02-08:00
Duration: 3 min 34 s
Channel: Meredith Moody
Description: The sum of a number and its square is 182 . Find the number(s).

The sum of a number and its square is 182 . Find the...

Key Concepts

Quadratic Formula

The quadratic formula, given as x = (-b ± ?(b² - 4ac)) / (2a), is a universal method for solving any quadratic equation, especially when the equation does not factor easily. It leverages the coefficients of the quadratic equation to determine both real and complex solutions.

Factoring

Factoring is an algebraic method used to simplify and solve quadratic equations by expressing the quadratic as a product of binomials. This technique is particularly useful when the equation factors neatly into terms that provide easy identification of the solutions.

Quadratic Equations

A quadratic equation is a second-degree polynomial equation typically written in the form ax² + bx + c = 0. Such equations are fundamental in algebra and can describe a variety of phenomena, with solutions representing the points where a corresponding quadratic function intersects the x-axis.

Translating Word Problems into Equations

This concept involves converting a verbal description into an algebraic equation. In problems where a number's properties are described, key phrases must be translated into mathematical expressions, thereby allowing the use of algebraic techniques to solve for unknowns.

Transcript

00:01 In order to solve this problem, i'm going to rewrite this equation in quadratic form.

00:08 And so i'm going to subtract 182 from both sides.

00:13 And that's going to leave me with my number and then the square of that number minus 182 equals 0.

00:24 Now i'm going to rewrite this in the standard quadratic form.

00:27 So the square of the number plus the number, minus 182.

00:36 Let's try that again.

00:41 82 equals 0.

00:43 Now i'm going to factor this in order to solve for my number.

00:49 So n squared plus n minus 182 equals zero.

00:54 I'm going to draw my parentheses for my factoring.

00:59 And because my c term is negative, i know that my factors are going to have one addition and one subtraction.

01:05 And because my a term has a coefficient of one, i know that i'm simply going to have n times n.

01:13 So now all i have to do is factor 182 and figure out which combination is going to give me a difference of positive one.

01:20 So this is even.

01:22 So then we get 91 and 91 is divisible by seven and it's seven times 13 so i have two times seven times 13 well i see that two times seven is 14 and so if i then say okay 14 times 13 is 182 but 14 minus 13 is one that is the combination that i need and since i'm supposed to be left with positive one my 14 needs to go in the addition factor and my 13 needs to go in the subtraction factor.

02:00 So now all i'm going to do is set these equal to zero separately so that i can solve for my value...

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