Determine the equation for the parabola with roots 2 + √5 and 2 − √5 and passing through the point (2,5).
with those roots,
y = a(x-(2 + √5))(x-(2 - √5)) = a(x^2-4x-1)
So now use the given point to find a.
To determine the equation for the parabola, we need to apply the concept of the quadratic equation.
Step 1: Find the equation in vertex form
The parabola is given with roots 2 + √5 and 2 − √5. This means that the equation will have the form:
(x - (2 + √5))(x - (2 - √5)) = 0
Step 2: Simplify
Expanding the equation in Step 1, we get:
(x - 2 - √5)(x - 2 + √5) = 0
(x - 2)^2 - (√5)^2 = 0
(x - 2)^2 - 5 = 0
(x - 2)^2 = 5
Step 3: Convert to standard form
Converting the equation in Step 2 to standard form, we get:
x^2 - 4x + 4 - 5 = 0
x^2 - 4x - 1 = 0
Therefore, the equation of the parabola with roots 2 + √5 and 2 − √5, and passing through the point (2,5) is:
y = x^2 - 4x - 1