Question

1. Find values of the parameter m so that the graph of the quadratic function f given by f(x)=x+x+1 and the graph of the line whose equation is given by y=mx​

Answer 1

We have to find the parameter m so that the graph of the quadratic function f given by f(x) = x² + x + 1 and the graph of the line whose equation is given by y = mx intersect each other.

f(x) = y = x² + x + 1 ...(1)

and y = mx ...(2)

putting equation (2) in equation (1) we get,

mx = x² + x + 1

⇒x² + (m - 1)x + 1 = 0

it will be defined only when, Discriminant ≥ 0

so, D = (m - 1)² - 4 × 1 × 1 ≥ 0

⇒m² - 2m + 1 - 4 ≥ 0

⇒m² - 2m - 3 ≥ 0

⇒m² - 3m + m - 3 ≥ 0

⇒(m - 3)(m + 1) ≥ 0

⇒m ≥ 3 or, m ≤ -1

Therefore both quadratic and straight line intersect each other only when m ∈ [3, ∞) U (-∞ , -1 ]