To begin solving the quadratic equation represented by y = 3x² + 4x + 11, we will identify the coefficients in the standard form of a quadratic equation, which is:
ax² + bx + c = 0
For our equation, we first set y to 0:
0 = 3x² + 4x + 11
Here, the coefficients are:
- a = 3
- b = 4
- c = 11
Next, we can apply the quadratic formula, which is expressed as:
x = (-b ± √(b² – 4ac)) / (2a)
1. **Calculate the Discriminant:** Begin by calculating the discriminant (b² – 4ac):
Discriminant = 4² – 4 × 3 × 11 = 16 – 132 = -116
2. **Determine the Nature of the Roots:** Since the discriminant is negative (-116), this indicates that there are no real solutions to the equation; instead, we will have two complex solutions.
3. **Plug the values into the Quadratic Formula:** Now, substitute the values of a, b, and the discriminant into the quadratic formula:
x = (-4 ± √(-116)) / (2 × 3)
4. **Simplify the Square Root of the Negative Discriminant:** The square root of -116 can be simplified as follows:
√(-116) = √(116) * i = 2√29 * i
5. **Final Calculation of x:** Now, substituting back, we have:
x = (-4 ± 2√29 * i) / 6
This simplifies to:
x = -2/3 ± (√29/3 * i)
Thus, the solutions to the equation y = 3x² + 4x + 11 are:
x = -2/3 + (√29/3 * i) and x = -2/3 – (√29/3 * i). Both are complex numbers, confirming the absence of real intersections with the x-axis.
In summary, by applying the quadratic formula to our equation, we determined that it yields complex solutions, indicating the curve does not cross the x-axis.