Determine the

A quadratic function with zeros at  a  and  b  is:

y  =  (x - a)(x - b)

Then we can multiply out the right side of that equation, or "FOIL" it, to get:

y  =  x² - bx - ax + ab

And then continue to simplify the right side until it looks like a nice quadratic equation.

Using this "template", we can find a solution to each of these problems.

a)    y   =   (x - 3)(x - 5)   =   x² - 5x - 3x + 15   =   x² - 8x + 15

b)    y   =   (x + 4)(x - 3)   =   x² - 3x + 4x - 12   =   x² + x - 12

c)    y   =   \((x-\frac12)(x-\frac23)\ =\ x^2-\frac23x-\frac12x+\frac13\ =\ x^2-\frac76x+\frac13\)

d)    In this case, the one root is  2 + √3  and the other is  2 - √3 , so the quadratic equation is:

        y  =  ( x - (2 + √3) )( x - (2 - √3) )

        y  =  ( x - 2 - √3 )( x - 2 + √3 )

        y  =  x² - 4x + 1

Can you figure out the last one? It might be the trickiest one, so if you need more help on it just ask!

Here is a graph to check the answers: https://www.desmos.com/calculator/lv66bdc1jy