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Math Help - Need help with roots of a equation

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    L44
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    Need help with roots of a equation

    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
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    Re: Need help with roots of a equation

    Quote Originally Posted by L44 View Post
    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}.
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    L44
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    Re: Need help with roots of a equation

    Yes I know that part. need to find the equation for the (α -β)2
    roots
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    Re: Need help with roots of a equation

    Quote Originally Posted by L44 View Post
    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?
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    Re: Need help with roots of a equation

    \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 = \, ?
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    L44
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    Re: Need help with roots of a equation

    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
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    Re: Need help with roots of a equation

    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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