How are quadratic equations solved ? Please explain with the example: u^2 + 6u = 27.

We have to solve the equation u^2 + 6u =
27.

First bring all the terms to one
side

=> u^2 + 6u - 27 =
0

One method of finding the roots which works when the
roots are integers or fractions is to split the coefficient of u into two parts which
add up to 6 and the product of which is -27. We can achieve this with 9 and
-3.

=> u^2 + 9u - 3u - 27 =
0

remove the common factors of the first two terms and the
last two terms.

=> u(u + 9) - 3(u +
9)

factorize again

=>(u
- 3)(u + 9) = 0

u - 3 = 0 => u =
3

and u + 9 = 0 => u =
-9

This gives us the roots of the equation as
3 and -9.