How do you solve the equation 3x^2 + 2x + 7 = 0?

To solve the quadratic equation 3x² + 2x + 7 = 0, we can apply the quadratic formula, which is given by:

x = (-b ± √(b² – 4ac)) / (2a)

In this equation:

• a = 3
• b = 2
• c = 7

Firstly, we will calculate the discriminant (the value under the square root in the formula), which is:

D = b² – 4ac

Substituting our values into the discriminant formula:

D = 2² – 4(3)(7) = 4 – 84 = -80

Since the discriminant is negative (-80), this means that the equation has no real solutions; instead, it has two complex (or imaginary) solutions.

To find the complex solutions, we plug the values back into the quadratic formula:

Thus, we have:

x = [−2 ± √(−80)] / (2 * 3)

This simplifies to:

x = [−2 ± √(80)i] / 6, where i is the imaginary unit.

This can be further simplified as:

x = [−1 ± (2/3)√(80)i]

Finally, since √(80) can be simplified to 4√(5), the solutions can be expressed as:

x = [−1 ± (4/3)√(5)i]

Therefore, the equation 3x² + 2x + 7 = 0 has two complex solutions:

• x = −1 + (4/3)√(5)i
• x = −1 − (4/3)√(5)i