What are the solutions to the equation 4x^2 + 3x – 24 = 0?

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To find the solutions to the quadratic equation 4x² + 3x – 24 = 0, we can use the quadratic formula:

x = (-b ± √(b² – 4ac)) / (2a)

Here, the coefficients are:

  • a = 4
  • b = 3
  • c = -24

Now, we need to calculate the discriminant, which is the part under the square root in the quadratic formula:

b² – 4ac = (3)² – 4(4)(-24)

Calculating that gives us:

9 + 384 = 393

Since the discriminant is positive (393 > 0), we can conclude that there are two real and distinct solutions.

Now, substituting the values of a, b, and the discriminant back into the quadratic formula:

1. First solution:

x₁ = [−3 + √393] / (2 * 4)

x₁ = [−3 + 19.84] / 8

x₁ ≈ 2.23

2. Second solution:

x₂ = [−3 − √393] / (2 * 4)

x₂ = [−3 – 19.84] / 8

x₂ ≈ -2.73

So, the solutions to the equation 4x² + 3x – 24 = 0 are approximately:

  • x ≈ 2.23
  • x ≈ -2.73

In summary, by applying the quadratic formula to the given equation, we found two distinct real solutions that can be of great interest to anyone seeking to solve quadratic equations!

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