Product of the Roots

How is 112 done?

 

112.

show that products of the roots of a quadratic equation is c/a

Let the quadratic equation be ax^2+bx+c=0, where a is not equal to 0

since a is not equal to 0, we have

a(x^2+(b/a)x+c/a)=0

=>x^2+(b/a)x+c/a=0......eq.1

now adding b^2/(4a^2) on both sides eq.1, we have

......move c/a to the right

.........complete square on left side



.........solve for x





now, roots (values) of the quadratic equation are

and

required product of roots will be




=.......this is difference of squares, so we have

=

= ....expand second term

=....simplify

=

=

 

Another job well-done. The "required product of roots" is just the quadratic formula separated into two parts and multiplied. Yes?

 

Here's another method: Any quadratic equation can be factored as a(x- x_1)(x- x_2)= 0 where x_1 and x_2 are the roots (a, x_1, and x_2 are not necessarily integers or even real numbers).

So ax^2- a(x_1+x_2)x+ ax_1x_2= 0.

Assuming that the original quadratic equation was ax^2+ bx+ c= 0 (which was NOT actually said) then c/a= (ax-1x-2)/a= x_1x_2, the product of the roots.

Notice also that -b/a= -(-a(x_1+ x_2))/a= x_1+ x_2, the sum of the roots (that was number 111).

(How many times are you going to post these same problems? I count four times so far.)

 

Again, I simply posted the sage attachment requesting help with different problems. Why are you making life difficult here?