WAEC 2014 · Paper 2 · Q2

Polynomials & quadratic roots

(a)
If $2x^2 + 3x + 3 = kx - k$ has real roots, find the range of values of $k$ .
Show the answer
$k \le -1$ or $k \ge 15$

Worked solution (try it first)

Get 0 on one side: $2x^2 + 3x - kx + 3 + k = 0$ .
Group the $x$ terms: $2x^2 + (3 - k)x + (3 + k) = 0$ .
So $a = 2$ , $b = 3 - k$ , $c = 3 + k$ .
Real roots need $b^2 - 4ac \ge 0$ : $(3 - k)^2 - 8(3 + k) \ge 0$ .
Expand: $9 - 6k + k^2 - 24 - 8k \ge 0$ .
Collect terms: $k^2 - 14k - 15 \ge 0$ .
Factorise: $(k - 15)(k + 1) \ge 0$ , with roots $k = -1$ and $k = 15$ .
" $\ge 0$ " is outside the roots: $k \le -1$ or $k \ge 15$ .

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