Want to understand why did we complete the square here. when we coudlve simplified it with x + 4x. For finding the minimum value do we always need to complete the square or am i missing something?
What do you mean?
we coudlve factored it no? x square + x + 4x + + 4
Try your method for, say, x^2 + 12x + 40.
You can try factoring, but you may have to use completing the square to find the vertex of a parabola.
For factoring, you must set the equation equal to zero and use factoring to find the roots. The midpoint of the roots is the x-coordinate of the vertex. If there is only one root, then you have found the x-coordinate of the vertex. If there are no roots, then you have to use completing the square.
In his example, we set x^2+5x+4 equal to 0 and factor it.
x^2+5x+4 = (x+1)(x+4) = 0 → x= -1 or x = -4
The x-coordinate of the vertex is the midpoint → [-1 + (-4)]/2 = -2.5.
Plug in x = -2.5 to find the y-coordinate of the vertex.
Hope this helps!