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June 2016 p11 q6b
290
The function \(f\) is defined for \(x \in \mathbb{R}\) by \(f(x) = x^2 + ax + b\), where \(a\) and \(b\) are constants. The solutions of the equation \(f(x) = 0\) are \(x = 1\) and \(x = 9\). Find:
the values of \(a\) and \(b\),
the coordinates of the vertex of the curve \(y = f(x)\).
Solution
Given the function \(f(x) = x^2 + ax + b\) with roots \(x = 1\) and \(x = 9\), we can express it as:
\(f(x) = (x - 1)(x - 9)\)
Expanding this, we get:
\(f(x) = x^2 - 10x + 9\)
Thus, \(a = -10\) and \(b = 9\).
To find the vertex, use the vertex formula \(x = \frac{-b}{2a}\) or symmetry: