How many roots do the following equations have: 12x² + 25x + 5 = 0 and x³ = 0?

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To determine the number of roots for the given equations, we will analyze each one separately.

1. Quadratic Equation: 12x² + 25x + 5 = 0

The equation is a quadratic equation of the form ax² + bx + c = 0, where:

  • a = 12
  • b = 25
  • c = 5

To find the number of roots, we will use the discriminant formula, which is given by:

D = b² - 4ac

Calculating the discriminant:

D = (25)² - 4(12)(5) = 625 - 240 = 385

Since the discriminant (D) is positive (385 > 0), this means that the quadratic equation has:

  • Two distinct real roots.

2. Cubic Equation: x³ = 0

The cubic equation can be factored as:

x³ = x 	imes x 	imes x = 0

This means:

  • x = 0 is the only root.

However, it is important to note that the root x = 0 has a multiplicity of 3. Therefore, we can conclude that while there is one unique root, it counts as:

  • Three roots (accounting for its multiplicity).

Summary

In conclusion:

  • The equation 12x² + 25x + 5 = 0 has two distinct real roots.
  • The equation x³ = 0 has one unique root with a multiplicity of three.

This analysis shows that the first equation has a different nature of roots compared to the second, with the quadratic yielding distinct roots and the cubic focusing on a single root with repetition.

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