# write down three quadratic equations with the following root and prove the roots are in fact the values of x for which y is equal to zero with the following root and prove the roots are in fact...

write down three quadratic equations with the following root and prove the roots are in fact the values of x for which y is equal to zero

Equation equals

two roots both equal to zero

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If you want to find a quadratic equation with the roots r and s, just plug r and s into:

y=(x-r)(x-s)

and distribute.

or more generally, pick any number a (other than 0) and plug r and s into:

y=a(x-r)(x-s)

So, for example, if I wanted a quadratic with the roots 1 and -2, one possibility is:

y=(x-1)(x+2) = x^2 +x - 2

another is:

y=2(x-1)(x +2) = 2x^2 + 2x - 4

So, in your case, we want the roots to be 0 and 0. We want three different equations, so we are going to use three different possibilites for a. The easiest is just a = 1,2,3.

So:

y=(x-0)(x-0) = x^2

y=2(x-0)(x-0) = 2x^2

y=3(x-0)(x-0) = 3x^2

Now, we want to check that these really do work.

So, plug x=0 into each of the following:

`y=x^2`

`y=2x^2`

`y=3x^2`

`y=(0)^2 = 0`

`y=2(0)^2=0`

`y=3(0)^2 = 0`

Sure enough, when we plug in x = 0, we get y=0

So 0 really is the root (double root in fact) of these quadratics