To solve the quadratic equation 3x² + 2x + 7 = 0, we will use the quadratic formula:
x = (-b ± √(b² – 4ac)) / (2a)
In our equation, the coefficients are:
- a = 3
- b = 2
- c = 7
Now, we will first calculate the discriminant, which is b² – 4ac:
- b² = 2² = 4
- 4ac = 4 * 3 * 7 = 84
So, the discriminant is:
4 – 84 = -80
Since the discriminant is negative, this indicates that the quadratic equation has no real solutions, and the solutions will be complex numbers.
Next, we calculate the solutions:
x = (-2 ± √(-80)) / (2 * 3)
To simplify the square root of -80, we can express it as:
√(-80) = √(80) * √(-1) = √(16 * 5) * i = 4√5 * i
Now, substituting back into the equation:
x = (-2 ± 4√5 * i) / 6
Breaking this down:
x = -1/3 ± (2/3)√5 * i
To express the solutions more clearly, we can evaluate the real and imaginary parts:
- x₁ = -1/3 + (2/3)√5 * i
- x₂ = -1/3 – (2/3)√5 * i
Thus, the final solutions to the equation 3x² + 2x + 7 = 0 are:
- x₁ ≈ -0.33 + 1.49i
- x₂ ≈ -0.33 – 1.49i
Since we are looking for the nearest hundredth, the answers are:
- x₁ ≈ -0.33 + 1.49i
- x₂ ≈ -0.33 – 1.49i
The solutions are rounded to the nearest hundredth for the real part, but the imaginary part remains in its approximate form because we can’t round imaginary components like real numbers.