To find the roots of the quadratic equation y = x² + 3x + 10, we will use the quadratic formula. The quadratic formula is given by:
x = (-b ± √(b² – 4ac)) / 2a
In this equation, a, b, and c are the coefficients from the standard form of a quadratic equation ax² + bx + c = 0. For our equation:
- a = 1
- b = 3
- c = 10
We will first need to calculate the discriminant (b² – 4ac):
- b² = 3² = 9
- 4ac = 4 * 1 * 10 = 40
Now, substitute these values into the discriminant:
Discriminant = 9 – 40 = -31
Since the discriminant is negative (-31), it indicates that the roots of the equation are not real numbers but complex numbers. We can proceed to find the complex roots using the quadratic formula:
x = (-3 ± √(-31)) / 2(1)
Now, we can simplify this expression:
Since √(-31) = i√31, where i is the imaginary unit, we have:
x = (-3 ± i√31) / 2
This results in two complex roots:
- Root 1: x = (-3 + i√31) / 2
- Root 2: x = (-3 – i√31) / 2
In conclusion, the roots of the equation y = x² + 3x + 10 are complex numbers and are represented as:
- x = (-3 + i√31) / 2
- x = (-3 – i√31) / 2