# 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 zerowith 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

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