Question

What is the discriminant of `x^2-4x+4=0` and what does that mean?

Answer 1

The discriminant is zero. It tells you that there are two identical real roots to the equation.

Explanation:

If you have a quadratic equation of the form

`ax^2+bx+c=0`

The solution is

`x = (-b±sqrt(b^2-4ac))/(2a)`

The discriminant `Δ` is `b^2 -4ac`.

The discriminant "discriminates" the nature of the roots.

There are three possibilities.

• If `Δ > 0`, there are two separate real roots.
• If `Δ = 0`, there are two identical real roots.
• If `Δ <0`, there are no real roots, but there are two complex roots.

Your equation is

`x^2 -4x + 4 = 0`

`Δ = b^2 – 4ac = (-4)^2 -4×1×4 = 16 - 16 = 0`

This tells you that there are two identical real roots.

We can see this if we solve the equation by factoring.

`x^2 -4x + 4 = 0`

`(x-2)(x-2) = 0`

`x-2 = 0` or `x-2 = 0`

`x = 2` or `x= 2`

There are two identical real roots to the equation.

Answer 2

The discriminant `Delta` characterize your solutions.

Explanation:

The discriminant `Delta` is a number that allows you to find out what type of solutions your equation will have.

1] If the discriminant is positive you'll have 2 separate real solutions `x_1!=x_2`;

2] If the discriminant is equal to zero you'll have 2 coincident real solutions, `x_1=x_2` (=two equal numbers...I know it is weird but do not worry);

3] If the discriminant is negative you'll have two complex solutions (in this case, at least for now, you stop and say that there will not be REAL solutions).

The discriminant is given as:
`color(red)(Delta=b^2-4ac)` where the letters can be found writing your equation in the general form:
`ax^2+bx+c=0` or in your case:
`x^2-4x+4=0`
so:
`a=1`
`b=-4`
`c=4`
and `Delta=(-4)^2-4(1*4)=16-16=0`
So you have case 2] two coincident solutions (if you solve your equation you'll find that it is satisfied by `x_1=x_2=2`).