What is the discriminant of $x^2 + 25 = 0$ and what does that mean?

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Answer 1 · 2015-07-18T19:44:08

$x^2+25 = 0$ has discriminant $-100 = -10^2$

Since this is negative the equation has no real roots. Since it is negative of a perfect square it has rational complex roots.

$x^2+25$ is in the form $ax^2+bx+c$ , with $a=1$ , $b=0$ and $c=25$ .

This has discriminant $Delta$ given by the formula:

$Delta = b^2-4ac = 0^2 - (4xx1xx25) = -100 = -10^2$

Since $Delta < 0$ the equation $x^2+25 = 0$ has no real roots. It has a pair of distinct complex conjugate roots, namely $+-5i$

The discriminant $Delta$ is the part under the square root in the quadratic formula for roots of $ax^2+bx+c = 0$ ...

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

So if $Delta > 0$ the equation has two distinct real roots.

If $Delta = 0$ the equation has one repeated real root.

If $Delta < 0$ the equation has no real roots, but two distinct complex roots.

In our case the formula gives:

$x = (-0 +-10i)/2 = +-5i$

What is the discriminant of x^2 + 25 = 0x^2 + 25 = 0 and what does that mean?