Without solving the equation, find the value of the roots

[BOAS]

Homework Statement

23 - 5x - 4x² = 0

find ($\alpha$ - $\beta$)²

Homework Equations

In previous parts of the question I've calculated $\alpha$ + $\beta$, $\alpha$$\beta$, 1/$\alpha$ + 1/$\beta$ and ($\alpha$+1)($\beta$+1) but I can't think of any rules I know to help me solve the problem.

The Attempt at a Solution

Expanding ($\alpha$ - $\beta$)²

gives

$\alpha$²+$\beta$² -2$\alpha$$\beta$

But I don't know what I can do with this info.

I'd appreciate a nudge in the right direction!

Thanks

 

[Office_Shredder]

You can write your expression fairly simply in terms of $\alpha+\beta$ and $\alpha \beta$ (from which you can get the final answer according to your part 2)

 

[Mark44]

Homework Statement

23 - 5x - 4x² = 0

find ($\alpha$ - $\beta$)²

Homework Equations

In previous parts of the question I've calculated $\alpha$ + $\beta$, $\alpha$$\beta$, 1/$\alpha$ + 1/$\beta$ and ($\alpha$+1)($\beta$+1) but I can't think of any rules I know to help me solve the problem.

The Attempt at a Solution

Expanding ($\alpha$ - $\beta$)²

gives

$\alpha$²+$\beta$² -2$\alpha$$\beta$

But I don't know what I can do with this info.

I'd appreciate a nudge in the right direction!

Thanks

$(\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta$
You said that you have already calculated $\alpha + \beta$ and $\alpha\beta$.

 

[BOAS]

$(\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta$
You said that you have already calculated $\alpha + \beta$ and $\alpha\beta$.

Gah, I should have seen that.

Thanks!