Need help with roots of a equation

**L44** · November 18th 2012, 11:39 AM

Question :

Give that α & β are roots of the equation 2x²+ 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

**Plato** · November 18th 2012, 11:49 AM

Originally Posted by L44

Give that α & β are roots of the equation 2x²+ 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

In $ax^2+bx+c$ the sum of the roots is $-\frac{b}{a}$ and the products of the roots is $\frac{c}{a}$ .

**L44** · November 18th 2012, 11:55 AM

Yes I know that part. need to find the equation for the (α -β)2
roots

**Plato** · November 18th 2012, 12:19 PM

Originally Posted by L44

Yes I know that part. need to find the equation for the (α -β)2
roots

Well $\alpha^2+2\alpha\beta+\beta^2=\frac{25}{4}$ Why?

Well $\alpha^2-2\alpha\beta+\beta^2=\frac{25}{4}-4\alpha\beta$ Why?

Can you finish?

**skeeter** · November 18th 2012, 12:19 PM

$\alpha \beta = \frac{c}{a}$ , $\alpha + \beta = -\frac{b}{a}$

$(\alpha + \beta)^2 = \alpha^2 + 2\alpha\beta + \beta^2 = \frac{b^2}{a^2}$

$(\alpha - \beta)^2 = \alpha^2 - 2\alpha\beta + \beta^2 = \, ?$

**L44** · November 18th 2012, 12:24 PM

I have that part the answer I got was 20 1/4 but is the working of the sum of the roots part (α -β)^2 where I have (α -β) + (α -β) unable to figure out the value

**Soroban** · November 18th 2012, 01:36 PM

Hello, L44!

$\text{Given that }\alpha\text{ and }\beta\text{ are roots of the equation }2x^2 + 5x - 7 \:=\:0$

$\text{Find, without solving the equation, the value of }(\alpha -\beta).$

Need to calculate the sum of the roots and the product of the roots

You said that you already know that: . $\begin{Bmatrix}\alpha + \beta &=& \text{-}\frac{5}{2} & [1] \\ \\[-3mm] \alpha\beta &=& \text{-}\frac{7}{2} & [2] \end{Bmatrix}$

Square [1]: . $(\alpha+\beta)^2 \:=\:\left(\text{-}\tfrac{5}{2}\right)^2 \quad\Rightarrow\quad \alpha^2 + 2\alpha\beta + \beta^2 \:=\:\tfrac{25}{4}\;\;[3]$

-4 times [2]: . $\text{-}4(\alpha\beta) \:=\:\text{-}4\left(\text{-}\tfrac{7}{2}\right) \quad\Rightarrow\qquad -4\alpha\beta \:=\:14 \quad [4]$

Add [3] and [4]: . $\alpha^2 - 2\alpha\beta + \beta^2 \:=\:\tfrac{25}{4} + 14 \quad\Rightarrow\quad (\alpha - \beta)^2 \:=\:\tfrac{81}{4}$

Therefore: . $\alpha - \beta \:=\:\tfrac{9}{2}$

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