Question: If the roots of the equation $x^{2} - 2ax + a^{2} + a - 3 = 0$ are real and less than 3, then...

Question

If the roots of the equation $x^{2} - 2ax + a^{2} + a - 3 = 0$ are real and less than 3, then

Answer 1

Given equation is $x^{2} - 2ax + a^{2} + a - 3 = 0$

If roots are real, then $D \geq 0$

⇒ $4a^{2} - 4(a^{2} + a - 3) \geq 0$ ⇒ $- a + 3 \geq 0$ ⇒ $a - 3 \leq 0$ ⇒ $a \leq 3$

As roots are less than 3, hence $f(3) > 0$

$9 - 6a + a^{2} + a - 3 > 0$ ⇒ $a^{2} - 5a + 6 > 0$ ⇒ $(a - 2)(a - 3) > 0$ ⇒ $a < 2,a > 3$ . Hence a < 2 satisfy all the conditions.

Question: If the roots of the equation \(x^{2} - 2ax + a^{2} + a - 3 = 0\) are real and less than 3, then...