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 ) Question from sadiya, a student: Find the quadratic equation whose roots are -4 and 7

Suppose you were given the quadratic equation

$x^2 + 3x - 10 = 0$

and you were asked to find the roots. I expect you would factor the quadratic to get

$(x - 2)(x + 5) = 0$

and then conclude that the roots are $2$ and $-5.$

If you know the roots are $2$ and $-5$ can you trace the steps backwards to obtain the quadratic equation

$x^2 + 3x - 10 = 0?$

Is this the only quadratic equation with roots $2$ and $-5?$ What about

$7(x^2 + 3x - 10) = 0$

or

$-8(x^2 + 3x - 10) = 0$

or

$k(x^2 + 3x - 10) = 0$

where $k$ is any constant?

Penny

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