Question

Form an equation whose roots are 1/2 and -1/3.

Answer 1

Well a root means that y will equal zero when x is that root number. we are given the roots. So when x = 1/2 or -1/3 then y = 0.

it has to be quadratic with an x^2 since it has 2 roots. So we will write basic formula

y = (x + a) (x + b)

in this basic quadratic the opposite of a and b will be roots. this means negative a and negative b are roots or -a and -b are roots.

so we can fill in our roots for a and b if we want but.remember we.have to use the opposite of.our roots because we.use -a and -b.

so let's fill in our opposite.roots for.a and.b

y = (x + (-1/2)) (x + (-(-1/3)))

since I have two negatives in front of the 1/3 it becomes positive.. So

y = (x - 1/2)(x + 1/3)

this is an equation with those roots. You can multiply it out if necessary.

y = x^2 - 1/6x - 1/6

I think that's what it comes out to.

Answer 2

==> When any factor of a number is zero, the number itself is zero.

==> When any factor of an expression is zero, the whole expression is zero.

So we can construct an equation that says an "expression is equal to zero",
if we make the expression out of 2 factors ... one that's zero when x=1/2,
and the other one that's zero when x= -1/3 .  Clever ? ! ?

OK.  One factor ==>   (x - 1/2) .   That's zero when  x= 1/2 .

The other factor ==>  (x + 1/3) .   That's zero when  x = -1/3 .

Multiply the factors, and you have the expression:  (x-1/2) (x+1/3) .

Proclaim that the expression is equal to zero:

                         (x - 1/2) · (x + 1/3)  =  0   .

Voila !  Now you have an equation that's only a true statement
IF  x = 1/2  OR  x = -1/3 .

In other words,  1/2 and -1/3 are the solutions, or roots, of the equation.

If U wanna FOIL it out and get another way to write it
without parentheses, go ahead:

                     x² - 1/6x - 1/6  =  0 

If you wanna multiply each side by 6 and get another way
to write it without fractions, go ahead:

                           6x² - x - 1  =  0