Question

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

Answer 1

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`

Answer 2

`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`

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

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

`" "64->64`