Question

If a number is added to its square, the result is 56. How do you find the number?

Answer 1

`color(blue)(a=-8)`

`color(blue)(a=7)`

Explanation:

Let the number be `a`:

`:.`

`a^2+a=56`

We now solve the equation:

`a^2+a-56=0`

Factor:

`(a+8)(a-7)=0=>a=-8 and a=7`

Answer 2

Call the number `n`, then we know `n+n^2=56`. We can write this as a quadratic equation in standard form: `n^2+n-56=0`. Using the quadratic formula, or by factorising, we find `n=-8` or `7`..

Explanation:

I normally use the quadratic formula: `n=(-b+-sqrt(b^2-4ac))/(2a)`

It's not hard to remember, and it always works.

For this particular one, though, I will factorise instead. I need two numbers that multiply to give `-56` and add to give `+1`. Two numbers that fit are `8` and `-7`.

So `(n+8)(N-7)=0`

If you expand that you will get back our quadratic.

The two 'roots' - the solutions of the quadratic - are the values of `n` that make each bracket go to zero.

Those roots are `-8` and `7`. Either value fulfils the criteria in the question.