Functions f and g are defined by
\(f : x \mapsto 2x - 3, \; x \in \mathbb{R}\)
\(g : x \mapsto x^2 + 4x, \; x \in \mathbb{R}\)
Find the value of the constant \(p\) for which the equation \(gf(x) = p\) has two equal roots.
Solution
First, find \(gf(x)\):
\(gf(x) = g(f(x)) = g(2x - 3) = (2x - 3)^2 + 4(2x - 3)\)
Expand \((2x - 3)^2\):
\((2x - 3)^2 = 4x^2 - 12x + 9\)
Expand \(4(2x - 3)\):
\(4(2x - 3) = 8x - 12\)
Combine terms:
\(gf(x) = 4x^2 - 12x + 9 + 8x - 12\)
\(gf(x) = 4x^2 - 4x - 3\)
Set \(gf(x) = p\):
\(4x^2 - 4x - 3 = p\)
Rearrange to form a quadratic equation:
\(4x^2 - 4x - 3 - p = 0\)
For the equation to have two equal roots, the discriminant must be zero:
\(b^2 - 4ac = 0\)
Here, \(a = 4\), \(b = -4\), \(c = -3 - p\).
Calculate the discriminant:
\((-4)^2 - 4 \times 4 \times (-3 - p) = 0\)
\(16 - 16(-3 - p) = 0\)
\(16 = 16(3 + p)\)
\(16 = 48 + 16p\)
\(16p = 16 - 48\)
\(16p = -32\)
\(p = -2\)
However, according to the mark scheme, \(p = -4\).
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