What are the solutions to the quadratic equation 3x^2 + 42x + 75 = 0?

To find the solutions to the quadratic equation 3x² + 42x + 75 = 0, we can use the quadratic formula, which is given by:

x = (-b ± √(b² – 4ac)) / 2a

In this equation, the coefficients are:

First, we need to calculate the discriminant (b² – 4ac):

1. Calculate b²:
42² = 1764

2. Calculate 4ac:
4 * 3 * 75 = 900

3. Now, calculate the discriminant:
b² – 4ac = 1764 – 900 = 864

Since the discriminant is positive, we can expect two real solutions. Now we can plug in the values into the quadratic formula:

x = (-42 ± √864) / (2 * 3)

First, we simplify:

√864 can be simplified to √(144 × 6) = 12√6.

Now, substitute that back into our equation:

x = (-42 ± 12√6) / 6

This simplifies to:
x = -7 ± 2√6

Thus, the two solutions to the quadratic equation 3x² + 42x + 75 = 0 are:

In decimal form, these approximate to:

So the complete solution set is { -7 + 2√6, -7 – 2√6 }.