To find the solutions to the quadratic equation 4x² + 22x + 36 = 0, we can use the quadratic formula:
x = (-b ± √(b² – 4ac)) / 2a
In our equation, the coefficients are:
- a = 4
- b = 22
- c = 36
First, we need to calculate the discriminant, which is b² – 4ac:
b² = 22² = 484
4ac = 4 × 4 × 36 = 576
Now, we can find the discriminant:
Discriminant = 484 – 576 = -92
Since the discriminant is negative, this indicates that the quadratic equation has no real solutions and instead has two complex solutions.
Next, we can substitute back into the quadratic formula:
x = (-22 ± √(-92)) / (2 × 4)
We can simplify this further:
x = (-22 ± √(92)i) / 8
Breaking that down gives us:
√92 = √(4 × 23) = 2√23, thus:
x = (-22 ± 2√23i) / 8
Now simplify:
x = -22/8 ± 2√23i/8
Which simplifies to:
x = -11/4 ± √23/4 i
Thus, the solutions to the quadratic equation 4x² + 22x + 36 = 0 are:
x = -11/4 + √23/4 i and x = -11/4 – √23/4 i