There are 3 main methods that are used to solve quadratics.
This method is best applied by strong students who can inspect quadratics effectively. When a quadratic is in the form
$$0=ax^2 + bx + c$$
We can apply some smart tricks to factor this expression into two linear factors.
Let us suppose that the roots are $r_1, r_2$.
So we know that
$$0=a(x-r_1)(x-r_2)$$
$$0=a(x^2 - (r_1 + r_2)x + r_1r_2)$$
$$0=ax^2 - a(r_1 + r_2)x + ar_1r_2$$
Since we know $a$, $b$, and $c$, we try to find numbers that multiply to $ac$ and add to $b$, which then helps us factor the quadratic.
The method of completing the square takes a quadratic and turns it into a perfect square trinomial, which is in the form
$$(x+k)^2 = x^2 + 2kx + k^2$$
This method turns a quadratic from standard form into vertex form.
$$y = ax^2 + bx + c$$
To do this method, we must make sure that the coefficient of $x^2 $ is 1, so we divide by $a$.
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