What does the discriminant of the quadratic equation 2x + 5x^2 + 1 represent, and how do you calculate it?

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The discriminant is an important concept in the context of quadratic equations, as it provides valuable information about the nature of the roots of the equation. For a standard quadratic equation in the form:

ax2 + bx + c = 0

the discriminant (denoted as D) is given by the formula:

D = b2 - 4ac

In your case, the quadratic equation is:

2x + 5x2 + 1 = 0

First, we need to rearrange this equation into the standard form:

5x2 + 2x + 1 = 0

Now, we can identify the coefficients:

  • a = 5
  • b = 2
  • c = 1

Next, we can substitute these values into the formula for the discriminant:

D = (2)2 - 4(5)(1)

Calculating this gives:

D = 4 - 20

Therefore:

D = -16

The value of the discriminant tells us about the nature of the roots of the quadratic equation:

  • If D > 0, the equation has two distinct real roots.
  • If D = 0, the equation has exactly one real root (also called a repeated root).
  • If D < 0, which is the case here, the equation has two complex (imaginary) roots.

Since the discriminant for the quadratic equation 5x2 + 2x + 1 is -16, we conclude that this quadratic equation has two complex roots.

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