Question

Write the equation of a parabola that has a complex root through 4 + 5i and goes through the point (2,87).

a) y = (x^2 + 8x + 25)
b) y = (x^2 - 8x + 25)
c) y = (x^2 + 8x + 69)
d) y = (x^2 - 8x + 69)

Answer 1

The equation of the parabola is y = (x-4)^2 + 25, therefore the correct answer is option a) y = (x^2 + 8x + 25).

Step-by-step explanation:

The equation of a parabola with a complex root can be written in the form y = ax^2 + bx + c, where a, b, and c are constants. Since the parabola passes through the point (2,87), we can substitute the values of x and y into the equation to obtain:

• 87 = a(2)^2 + b(2) + c
• 87 = 4a + 2b + c

With the given complex root 4 + 5i, we can use the fact that complex roots of quadratic equations occur in conjugate pairs. This means that if 4 + 5i is a root, then 4 - 5i must also be a root. Therefore, we know that the equation has the factors (x - (4 + 5i)) and (x - (4 - 5i)). Using the complex conjugate rule, we can determine that the equation is:

• y = (x - 4 - 5i)(x - 4 + 5i)
• y = (x - 4)^2 - (5i)^2
• y = (x - 4)^2 - 25i^2
• y = (x - 4)^2 - 25(-1)
• y = (x - 4)^2 + 25

So the correct equation of the parabola is: y = (x-4)^2 + 25. Therefore, the answer is option a) y = (x^2 + 8x + 25).