The square of a positive number is 56 more than the number itself. What is the number?

2 Answers
Jan 9, 2017

The number is #8#

Explanation:

We need to take this one phrase at a time to develop our equation.

First, the square of a positive number can be written as:

#x^2#

In mathematics the word "is"means "=" so we can now write:

#x^2 =#

and "56 more than the number itself" finishes the equation as:

#x^2 = 56 + x#

We can now transform this into a quadratic:

#x^2 - color(red)(56 - x) = 56 + x - color(red)(56 - x)#

#x^2 - x - 56 = 0#

We can now factor the quadratic:

#(x - 8)(x + 7) = 0#

Now we can solve each term for #0#

#x + 7 = 0#

#x + 7 - 7 = 0 - 7#

#x + 0 = -7#

#x = -7# - this cannot be the answer because the question asked for a positive integer.

#x - 8 = 0#

#x - 8 + 8 = 0 + 8#

#x - 0 = 8#

#x = 8#

The number is #8#

Jan 9, 2017

#8#

Explanation:

Let the unknown value be #x#

This is a quadratic in disguise.

#x^2=x+56" "=>" "x^2color(red)(-x)-56=0#

The #color(red)(x)# has the coefficient of -1. This means that the whole number factors of 56 have a difference of -1.

#sqrt(56)~~7.5#

Try #(-8)xx(+7) =-56 " and "7-8=-1# so we have found the factors

#x^2-x-56=(x-8)(x+7)=0#

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The question stipulates that the number is positive so we select #x=+8#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Check")#

#x^2=x+56" "->" "8^2->8+56#

#" "64->64#