Question

In quadratic equations, why do we solve by factoring, use square root property, do completing the square and do quadratic formulas. I can do them but dont understand why?

Answer 1

Factor: when the solutions to the equation are integers.
Complete the square: when you suspect that the solutions involve decimal numbers.
Quadratic formula: if you're lazy and don't feel like thinking.

Answer 2

...
The quadratic formula and other techniques for solving quadratic equations
ultimately help you solve for the “zeros of a quadratic polynomial”

For an equation ax^2 + bx + c = 0,
when you solve for it using the quadratic,
then you’ve found the “zeros” for f(x) = ax^2 + bx + c.
You’ve found the values for x that make the polynomial equal to zero.

Finding zeros for quadratic functions helps you
approximate zeros for higher degree polynomials or other more complicated functions.

I think quadratic equations first arose from physics.
If you throw a softball pitcher throw the ball underhanded,
you’ll notice the path of the softball is an upside down parabola.

If you watch 3 point basketball shots, they’re always parabolic paths.
All objects on Earth subject to gravity and in free fall move parabolically.

Solving quadratics lets you be able to answer questions like,
“If this book falls out of the 10th story floor, how long will it take to hit the ground?”
“If the basketball is shot at an angle of 45 degrees at 10 miles an hour, what is its maximum height?”

Now though, they’re used for much more than physics.
Anyone who uses math at all needs to understand them, from
biologists to economists to computer scientists.