Question

How do you use the discriminant to determine the nature of the solutions given `y = x^2 – 7x + 7`?

Answer 1

Have a look if it is not too confusing...!

Explanation:

The discriminant, or Delta `delta`, is a number obtained considering the numerical coefficients of your second degree equation, `a,b and c`.
In your case you have a function, that tells you all the possible solutions (combinations of `x` values) of an expression of the form `x^2-7x+7=?`.
For example, if you choose `x=1` you'll get: `1^2-7*1+7=1`
One interesting case is when:
`x^2-7x+7=0`. The problem here is that we do not know the `x` values that give me exactly zero as result!
But...no problem we use the discriminant to figure out what kind of solutions we'll get.
We consider the general form of our equation: `ax^2+bx+c=0`
where the corresponding numerical coefficients, in our equation, are:
`a=1`
`b=-7`
`c=7`
and the formula for the discriminant:
`color(red)(Delta=b^2-4ac)`
and we get:
`Delta=(-7)^2-4(1*7)=49-28=21`
Our Delta is bigger than zero so this implies that our equation will be satisfied by 2 real and distinct values of `x`;
If Delta was equal to zero we'd had 2 real coincident values (basically two equal numbers).
A value of Delta less than zero would imply no real solutions available for our equation.