Question

The standard form of the equation for a quadratic function is, with he vertex of the graph of the function at the point (p,q). the factored form is y=a(x-x1) (x-x2), where x1 and x2 are the x-intercepts of the graph. Express p and q in terms of x1 and x2 by expanding the factored for then completing the square. Please show work.

Answer 1

p = ½ (x₁ + x₂)

q = a (x₁x₂ − ¼ (x₁ + x₂)²)

Explanation:

y = a (x − x₁) (x − x₂)

Expand:

y = a (x² − x₁x − x₂x + x₁x₂)

y = a (x² − (x₁ + x₂)x + x₁x₂)

Distribute a to the first two terms:

y = a (x² − (x₁ + x₂)x) + ax₁x₂

Complete the square:

y = a (x² − (x₁ + x₂)x + ¼(x₁ + x₂)²) + ax₁x₂ − ¼ a(x₁ + x₂)²

y = a (x − ½ (x₁ + x₂))² + a (x₁x₂ − ¼ (x₁ + x₂)²)

Therefore:

p = ½ (x₁ + x₂)

q = a (x₁x₂ − ¼ (x₁ + x₂)²)