What is the discriminant of $3 x^{2} + 6 x + 5$ $3x^2 + 6x + 5$ and what does that mean?

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Answer 1 · 2015-07-27T12:28:05

For this quadratic, $Delta = -24$ , which means that the equation has no real solution, but that it does have two distinct complex ones.

For a quadratic equation written in general form

$ax^2 + bx + c = 0$ ,

the discriminant is defined as

$Delta = b^2 - 4 * a * c$

In your case, the quadratic looks like this

$3x^2 + 6x +5 = 0$ ,

which means that you have

${(a=3), (b=6), (c=5) :}$

The discriminant will thus be equal to

$Delta = 6^2 - 4 * 3 * 5$

$Delta = 36 - 60 = color(green)(-24)$

When $Delta<0$ , the equation has no real solutions. It does have two distinct complex solutions derived from the general form

$x_(1,2) = (-b +- sqrt(Delta))/(2a)$

which in this case becomes

$x_(1,2) = (-b +- isqrt(-Delta))/(2a)$ , when $Delta<0$ .

In your case, these two solutions are

$x_(1,2) = (-6 +- sqrt(-24))/(2 * 3)$

$x_(1,2) = (-6 +- isqrt(24))/6 = (-6 +- 2isqrt(6))/6 = {(x_1 = (-3 - isqrt(6))/3), (x_2 = (-3 + isqrt(6))/3) :}$

What is the discriminant of 3x^2 + 6x + 53x2+6x+53x^2 + 6x + 5 and what does that mean?