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Math Help - how to solve without solving

  1. #1
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    how to solve without solving

    Without solving the given equation, write an equation whose roots could be obtained by subtracting 1 from each root of x^2 - 4x - 6 = 0.

    Answer key: x^2 - 2x - 9

    How to solve this without solving? Is this asking for Sum of Roots and product of roots?

    S = ( ) + ( )
    P= ( ) ( )

    S = -b/a
    P = c/a

    I don't think the above 2 sum and product techniques work. What else can I do?

    Thanks.
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  2. #2
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    Quote Originally Posted by shenton View Post
    Without solving the given equation, write an equation whose roots could be obtained by subtracting 1 from each root of x^2 - 4x - 6 = 0.

    Answer key: x^2 - 2x - 9

    How to solve this without solving? Is this asking for Sum of Roots and product of roots?

    S = ( ) + ( )
    P= ( ) ( )

    S = -b/a
    P = c/a

    I don't think the above 2 sum and product techniques work. What else can I do?

    Thanks.
    Well, the linear term is easy: if we take 1 from each root then the sum of those roots will be decreased by 2:
    S' = S - 2
    S' = 4 - 2 = 2

    The product is a bit tougher and I don't know how to get this result without essentially solving the problem. What I did was this:
    The original expression is
    (x - a)(x - b)
    So we want
    (x - (a - 1))(x - (b - 1)) = x^2 - (a + b - 2)x + (ab - (a + b) + 1)

    So the last term will be:
    P' = P - S + 1
    P' = -6 - 4 + 1 = -9
    But I really don't know how to prove this result without working it out as above (which seems to be cheating to me.)

    -Dan
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  3. #3
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    Hello, shenton!

    Actually, you had it, Dan . . .


    Without solving the given equation, write an equation whose roots could be obtained
    by subtracting 1 from each root of x^2 - 4x - 6 \:= \:0

    Answer key: x^2 - 2x - 9

    Let a and b be the roots of the equation.
    Then: . a + b \,=\,4,\;ab\,=\,-6

    Let the roots of the new equation be: . \begin{array}{cc}A\:=\:a-1 \\ B\:=\:b-1\end{array}

    Then we have:
    . . A + B \;=\; (a-1)+(b-1) \;= \; (a + b) - 2 \;=\; 4 - 2\quad\Rightarrow\quad \boxed{A+B\:=\:2}
    AB \;= \; (a-1)(b-1) \;=\; ab - (a+b) + 1 \;=\; -6 - 4 + 1\quad\Rightarrow\quad \boxed{AB\:=\:-9}

    Therefore, the equation is: . \boxed{x^2 -2x - 9\:=\:0}

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