Maths graph help

From the function f(2a) is zero. Why do you think a root occurs at x=a?

You found the minimum, not the root. The quadratic is
x(x-2a)
The roots are obvious?

Factorize to get the roots (or use the quadratic formula), complete the square to get the stationary point.

Factorising you multiply the "numbers" to give -8a^2 and add to give 2a? So each factor must be something like
(x+3a)
... so what "numbers" work to give the -8a^2 and 2a?

Guessing two factors as
(x + 2a)(x+3a)
what do you get when you expand, how would you change the 2a and 3a to give your quadratic?

Alternative, factorise
x^2 + 2x - 8
Then modify slightly for your problem?

One vid
https://www.doubtnut.com/question-answer/factorise-x2-3ax-88a2-644442505
but just google?

Did you do the other two things, but just google for a factorise quadratic with ... Similarly it should be covered in your textbook.

Third time and last time, can you factorise
x^2 + 2x - 8
If so, how would you simply adapt to this quadratic

Video worked for me and google gave some hits as well. Factorising a quadratic where the linear and constant terms (in temrs of x) depend on another parameter (in this case a) come up from time to time, so it should be in there.