To find the roots of the quadratic equation y = x² + 3x + 10, we need to use the quadratic formula:
x = (-b ± √(b² – 4ac)) / 2a
In this equation, a is the coefficient of x², b is the coefficient of x, and c is the constant term. Here, we have:
- a = 1
- b = 3
- c = 10
First, we calculate the discriminant (b² – 4ac):
Discriminant = 3² – 4(1)(10) = 9 – 40 = -31
Since the discriminant is negative, this indicates that there are no real roots for the equation. Instead, we have two complex roots. We can proceed to find these complex roots:
x = (-3 ± √(-31)) / 2(1)
We know that √(-1) can be replaced with the imaginary unit i, so:
√(-31) = √31 * i
Substituting this back into our formula gives:
x = (-3 ± √31 * i) / 2
This can be written as:
x = -3/2 ± (√31/2)i
Thus, the roots of the equation y = x² + 3x + 10 are:
- x = -3/2 + (√31/2)i
- x = -3/2 – (√31/2)i