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

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 our case, the coefficients are:

• a = 3
• b = 42
• c = 75

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

Discriminant = b² – 4ac = 42² – 4 * 3 * 75

Calculating that gives us:

Discriminant = 1764 – 900 = 864

Since the discriminant is positive, we will have two distinct real solutions. Next, we substitute the values into the quadratic formula:

x = (-42 ± √864) / (2 * 3)

Now, simplify the square root of 864:

√864 = √(144 * 6) = 12√6

Substituting back, we have:

x = (-42 ± 12√6) / 6

Now, we can simplify this further:

x = -7 ± 2√6

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

1. x = -7 + 2√6
2. x = -7 – 2√6

These represent the values of x that satisfy the original quadratic equation.