Solutions of Quadratic Equations and Inequalities
There are 3 main methods that are used to solve quadratics.
Methods of Solving Quadratics
Factoring
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 $c$ and $b$, we look at the value of $c$ and the value of $b$ to try to find $r_1,r_2$ by factoring $c$ into two numbers that add to $b$
Completing the Square
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 $a$.
$$\frac{y}{a}= x^2 + \frac{b}{a}x + \frac{c}{a}$$
Now for the confusing part—for a perfect square trinomial, notice that the constant term is the coefficient of the $x$ term halved then squared.
$$(\frac{2k}{2})^2 = k^2$$
Therefore, to obtain a perfect square trinomial for $y$, we can add look at the coefficient of the $x$ term, and add a constant which has the same value as when said coefficient is halved then squared
$$\frac{y}{a} + \frac{b^2}{4a}= x^2 + \frac{b}{a}x + \frac{b^2}{4a}+ \frac{c}{a}$$
So now we can write the right hand side as
$$\frac{y}{a} + \frac{b^2}{4a^2}= (x+\frac{b}{2a})^2 + \frac{c}{a}$$
This allows us to isolate $x$, transitioning greatly into the next part
