First slide

Theory of equations

Question

If both the roots of the quadratic equation
$x^{2} - 2 kx + k^{2} + k - 5 = 0$
are less than 5, then k lies in the interval

Moderate

A

$(- \infty, 4)$
B

$[4, 5]$
C

$[5, 6]$
D

$(6, \infty)$

Solution

$x^{2} - 2 kx + k^{2} + k - 5 = 0 Roots are less than 5, D \geq 0 4 k^{2} - 4 (k^{2} + k - 5) \geq 0 . . . . . (i) \Rightarrow k \leq 5 \Rightarrow f (5) > 0 . . . . . (ii) \Rightarrow k \in (- \infty, 4) \cup (5, \infty); - (\frac{2 k}{2}) < 5 \Rightarrow k < 5 . . . . (iii) form (i), (ii) and (iii), k \in (- \infty, 4)$

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