# Thread: Find all values of parameter 'a' if?

1. ## Find all values of parameter 'a' if?

For what values of 'a' does the equation ax^2 - (a + 1)x + 3 = 0, have roots lying between 1 and 2?

2. The quadratic formula states for a quadratic equation $ax^2 + bx + c = 0$, the solution is
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

In your case, $a = a, b = -(a+1), c = 3$, so your solutions are
$x = \frac{a+1 \pm \sqrt{(a+1)^2 - 4(a)(3)}}{2a} = \frac{a+1 \pm \sqrt{a^2 - 10a + 1}}{2a}$

The small root is
$x_1 = \frac{a+1 - \sqrt{a^2 - 10a + 1}}{2a}$
while the larger root is
$x_2 = \frac{a+1 + \sqrt{a^2 - 10a + 1}}{2a}$

Your constraint say that $1 and $x_2<2$, so find the values of $a$ such that those two inequalities are satisfied.

Can you take it from there?