How can we find the solutions for the equation 2x² + 16x + 34 = 0?

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To solve the quadratic equation 2x² + 16x + 34 = 0, we can use the quadratic formula, which is:

x = (-b ± √(b² - 4ac)) / (2a)

In our equation, the coefficients are:

  • a = 2
  • b = 16
  • c = 34

Let’s plug these values into the quadratic formula step by step:

  1. Calculate the discriminant (b² – 4ac):
    2. b² = 16² = 256
4ac = 4 * 2 * 34 = 272
Discriminant = 256 - 272 = -16
  2. Since the discriminant is negative (-16), this indicates that the equation has no real solutions, but two complex solutions.
  3. Now, substituting the values into the quadratic formula:
    4. x = (-16 ± √(-16)) / (2 * 2)
√(-16) can be expressed as 4i (where i is the imaginary unit).
  4. Therefore, we can rewrite the equation as:
    5. x = (-16 ± 4i) / 4
  5. Simplifying gives:
    6. x = -4 ± i

In conclusion, the solutions for the equation 2x² + 16x + 34 = 0 are:

  • x = -4 + i
  • x = -4 – i

These solutions indicate that the graph of the quadratic equation does not intersect the x-axis. Instead, it opens upwards and lies entirely above the x-axis, reflecting its positive leading coefficient.

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