When solving algebraic equations, one question that often arises is whether the equation admits real solutions. Real solutions are those that can be expressed as real numbers, as opposed to complex numbers.

In the case of a quadratic equation of the form ax^2 + bx + c = 0, the solutions can be determined by using the discriminant, which is defined as b^2 – 4ac.

The Discriminant

The discriminant provides valuable information about the nature of the roots of the quadratic equation. It helps us determine whether the solutions are real or complex, and even provides information about the number of distinct solutions.

Let’s consider the equation 5x^2 + 2kx + 3 = 0 for different values of k. By calculating the discriminant, we can determine which values of k yield real solutions.

Calculating the Discriminant

The discriminant can be calculated using the formula b^2 – 4ac. In our equation, a = 5, b = 2k, and c = 3. Substituting these values into the formula, we get:

  • Discriminant = (2k)^2 – 4(5)(3)
  • Discriminant = 4k^2 – 60

Analyzing the Discriminant

Now, let’s analyze the discriminant to determine the values of k that result in real solutions.

  • If the discriminant is greater than 0 (D > 0), the equation admits two distinct real solutions.
  • If the discriminant is equal to 0 (D = 0), the equation admits two identical real solutions.
  • If the discriminant is less than 0 (D < 0), the equation admits two complex solutions with imaginary parts.

Applying the Discriminant to Find Real Solutions

Let’s apply this information to determine the values of k that yield real solutions.

If D > 0, we have:

  • 4k^2 – 60 > 0
  • 4k^2 > 60
  • k^2 > 15
  • k > √15 or k < -√15

If D = 0, we have:

  • 4k^2 – 60 = 0
  • 4k^2 = 60
  • k^2 = 15
  • k = √15 or k = -√15

If D < 0, we have:

  • 4k^2 – 60 < 0
  • 4k^2 < 60
  • k^2 < 15
  • -√15 < k < √15

Therefore, for the equation 5x^2 + 2kx + 3 = 0, real solutions exist when k > √15 or k < -√15.

Remember, this analysis holds for quadratic equations. For higher degree polynomials, the determination of real solutions becomes more complex, sometimes requiring numerical methods or advanced algebraic techniques.

By understanding the discriminant and performing the necessary calculations, we can confidently identify the values of k that lead to real solutions in the given equation.

Quest'articolo è stato scritto a titolo esclusivamente informativo e di divulgazione. Per esso non è possibile garantire che sia esente da errori o inesattezze, per cui l’amministratore di questo Sito non assume alcuna responsabilità come indicato nelle note legali pubblicate in Termini e Condizioni
Quanto è stato utile questo articolo?
0Vota per primo questo articolo!