Solve the following equations which have equal roots:
$x^{4} - 6 x^{3} + 12 x^{2} - 10 x + 3 = 0$ .

Open in App

Solution

Give equation, $x^{4} - 6 x^{3} + 12 x^{2} - 10 x + 3 = 0$

Consider $f (x) = x^{4} - 6 x^{3} + 12 x^{2} - 10 x + 3$

$∴ f^{'} (x) = 4 x^{3} - 18 x^{2} + 24 x - 10$

Now, HCF of $f (x)$ and $f^{'} (x)$ is $(x - 1)$ . Hence 1 is a double root of $f (x) = 0$

$f (x)$ can be factored as $(x - 1)^{2} (x^{2} - 4 x + 3) or (x - 1)^{2} (x - 1) (x - 3)$
$∴$ roots of the given equation are $1, 1, 1, 3$

Suggest Corrections

Similar questions

x^{4} - 2 x^{3} - 12 x^{2} + 10 x + 3 = 0

x^{4} - (a + b) x^{3} - a (a - b) x^{2} + a^{2} (a + b) x - a^{3} b = 0

x^{4} - 3 x^{2} + 2 = 0

f (x) = x^{4} - 6 x^{3} + 12 x^{2} - 8 x + 3

m

{12 x}^{2} + 4 (m + 1) x + 3 = 0

Join BYJU'S Learning Program