# How to use the discriminant to find out what type of solutions the equation has for -x^2 + 4x – 4 = 0?

May 25, 2015

The discriminant is used to determine whether there are any solutions at all. It is really just part of the quadratic formula.

$\sqrt{{b}^{2} - 4 a c}$

With $\left(-\right) {x}^{2} + \left(4\right) x + \left(- 4\right) = 0$:
$a = - 1$
$b = 4$
$c = - 4$

So we have:
$\sqrt{{4}^{2} - 4 \left(- 1\right) \left(- 4\right)} = \sqrt{16 - 16} = 0$
meaning that this equation has solutions. The solutions can be determined by factoring.

If you divide by -1, you can make this look nicer, and $\frac{0}{- 1} = 0$:
${x}^{2} - 4 x + 4 = 0$

Notice how if you divide the middle term by 2 and square it, you ger 4, so this is a perfect square. This is:

$\left(x - 2\right) \left(x - 2\right) = {\left(x - 2\right)}^{2} = 0$

Or, as it was written:
$- \left(x - 2\right) \left(x - 2\right) = - {\left(x - 2\right)}^{2} = 0$

$x = 2$ (multiplicity 2)