Question

How do you describe the nature of the roots of the equation `x^2-4x+4=0`?

Answer 1

We can factor the expression on the right as:

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

Therefore the two roots for this quadratic are the same: `x = 2`

Explanation:

We can also use the discriminate to show this same result:

The quadratic formula states:

For `ax^2 + bx + c = 0`, the values of `x` which are the solutions to the equation are given by:

`x = (-b +- sqrt(b^2 - 4ac))/(2a)`

The discriminate is the portion of the quadratic equation within the radical: `color(blue)(b)^2 - 4color(red)(a)color(green)(c)`

If the discriminate is:
- Positive, you will get two real solutions
- Zero you get just ONE solution
- Negative you get complex solutions

To find the discriminant for this problem substitute:

`color(red)(1)` for `color(red)(a)`

`color(blue)(-4)` for `color(blue)(b)`

`color(green)(4)` for `color(green)(c)`

`color(blue)((-4))^2 - (4 * color(red)(1) * color(green)(4)) =>`

`16 - 16 =>`

`0`

Because the discriminate is `0` you get just ONE solution.