First slide

Theory of equations

Question

If $x^{2} + px + q = 0$ is the quadratic equation whose roots are $a - 2$ and $b - 2$ where a and b are the roots of $x^{2} - 3 x + 1 = 0$ , then

Moderate

A

$p = 1, q = 5$
B

$p = 1, q = - 5$
C

$p = - 1, q = 1$
D

None of these

Solution

$a + b = 3, ab = 1 and - p = a - 2 + b - 2,$
$q = (a - 2) (b - 2)$
$\Rightarrow - p = a + b - 4, q = ab - 2 (a + b) + 4$
$\Rightarrow - p = 3 - 4 and q = 1 - 2 (3) + 4 \Rightarrow (p, q) = (1, - 1)$

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