Question :
Give that α & β are roots of the equation 2x^{2 }+ 5x - 7 = 0
Find without solving the equation the value of (α -β)^{2}
Need to calculate the sum of the root and the product of the roots
$\displaystyle \alpha \beta = \frac{c}{a}$ , $\displaystyle \alpha + \beta = -\frac{b}{a}$
$\displaystyle (\alpha + \beta)^2 = \alpha^2 + 2\alpha\beta + \beta^2 = \frac{b^2}{a^2}$
$\displaystyle (\alpha - \beta)^2 = \alpha^2 - 2\alpha\beta + \beta^2 = \, ?$
Hello, L44!
$\displaystyle \text{Given that }\alpha\text{ and }\beta\text{ are roots of the equation }2x^2 + 5x - 7 \:=\:0$
$\displaystyle \text{Find, without solving the equation, the value of }(\alpha -\beta).$
Need to calculate the sum of the roots and the product of the roots
You said that you already know that: .$\displaystyle \begin{Bmatrix}\alpha + \beta &=& \text{-}\frac{5}{2} & [1] \\ \\[-3mm] \alpha\beta &=& \text{-}\frac{7}{2} & [2] \end{Bmatrix}$
Square [1]: .$\displaystyle (\alpha+\beta)^2 \:=\:\left(\text{-}\tfrac{5}{2}\right)^2 \quad\Rightarrow\quad \alpha^2 + 2\alpha\beta + \beta^2 \:=\:\tfrac{25}{4}\;\;[3]$
-4 times [2]: .$\displaystyle \text{-}4(\alpha\beta) \:=\:\text{-}4\left(\text{-}\tfrac{7}{2}\right) \quad\Rightarrow\qquad -4\alpha\beta \:=\:14 \quad [4]$
Add [3] and [4]: .$\displaystyle \alpha^2 - 2\alpha\beta + \beta^2 \:=\:\tfrac{25}{4} + 14 \quad\Rightarrow\quad (\alpha - \beta)^2 \:=\:\tfrac{81}{4}$
Therefore: .$\displaystyle \alpha - \beta \:=\:\tfrac{9}{2}$