Final answer:
The equation of the parabola with x-intercepts at (1+√5, 0) and (1-√5, 0), passing through (4, 8), is y = 2x² - 4x - 8.
Step-by-step explanation:
To write the equation for the parabola that has x-intercepts at (1+√5, 0) and (1-√5, 0) and passes through the point (4, 8), we'll use the fact that the x-intercepts tell us about the factors of the quadratic equation of the parabola. The standard form of a parabola's equation is y = ax² + bx + c. Since the given x-intercepts are the roots of the equation, we can express it as:
y = a(x - (1 + √5))(x - (1 - √5))
Expanding this, we get:
y = a[(x - 1 - √5)(x - 1 + √5)]
y = a[(x - 1)² - (√5)²]
y = a[x² - 2x + 1 - 5]
y = a[x² - 2x - 4]
Next, use the point (4, 8) to solve for coefficient 'a'.
8 = a[4² - 2(4) - 4]
8 = a[16 - 8 - 4]
8 = a[4]
a = 2
Thus, the equation of the parabola is y = 2(x² - 2x - 4) or y = 2x² - 4x - 8.