First slide

Theory of equations

Question

If the roots of $x^{3} - 12 x^{2} + 39 x - 28 = 0$ are in A.P. , then their common difference, is

Moderate

A

$\pm 1$
B

$\pm 2$
C

$\pm 3$
D

$\pm 4$

Solution

Let $a - d, a, a + d$ be the roots of the equation $x^{3} - 12 x^{2} + 39 x - 28 = 0 .$ Then
$a - d + a + a + d = 12$ and, $(a - d) a (a + d) = 28$
$\Rightarrow 3 a = 12$ and $a (a^{2} - d^{2}) = 28$
$\Rightarrow a = 4$ and $d = \pm 3$

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