Take a look at the applet: Sine Functions
The general form of the quadratic formula is $f\left(x\right)=a{x}^{2}+bx+c$.
It is not immediately apparent how this function rule could be derived by transformation
of the basic power function $y={x}^{2}$. That makes it difficult to find the vertex and x-intersects of the corresponding
parabola.
Using a method called completing the square allows you to convert the function $f$ to the form: $f\left(x\right)=a{(x-p)}^{2}+q$ where $(p,q)$ are the coordinated of the vertex of the graph.
To do this conversion you use the following property:
${x}^{2}+2kx={(x+k)}^{2}-{k}^{2}$
Use the applet to check that $f\left(x\right)=2{x}^{2}-4x$ is the same function as $g\left(x\right)=2{(x-1)}^{2}-2$.
It is obviously very useful if by completing the square you can convert $f\left(x\right)=a{x}^{2}+bx+c$ to a form that allows you to immediately see the vertex and the axis of symmetry...
A long time ago, mathematicians derived the so-called quadratic formula.
This formula allows you to solve $a{x}^{2}+bx+c=0$ and thereby find the zeros of the quadratic equation. The general solution is:
$x=\frac{\mathrm{-}b+\sqrt{{b}^{2}-4ac}}{2a}\vee x=\frac{\mathrm{-}b-\sqrt{{b}^{2}-4ac}}{2a}$
Below you see a proof of the quadratic formula. This means you can show that the formula is always valid. To do so you need to solve $a{x}^{2}+bx+c=0$ in general terms by completing the square.
Assume that $a\ne 0$ (otherwise it would not be a quadratic equation!). You now divide by $a$ on both sides. This gives you:
${x}^{2}+\frac{b}{a}x+\frac{c}{a}=0$
Completing the square results in:
${(x+\frac{b}{2a})}^{2}-{\left(\frac{b}{2a}\right)}^{2}+\frac{c}{a}=0$ and ${(x+\frac{b}{2a})}^{2}={\left(\frac{b}{2a}\right)}^{2}-\frac{c}{a}=\frac{{b}^{2}-4ac}{4{a}^{2}}$
Taking the square root:
$x+\frac{b}{2a}=\pm \sqrt{\frac{{b}^{2}-4ac}{4{a}^{2}}}$
And now a few rearrangements:
$x=\mathrm{-}\frac{b}{2a}\pm \sqrt{\frac{{b}^{2}-4ac}{4{a}^{2}}}=\frac{\mathrm{-}b}{2a}\pm \frac{\sqrt{{b}^{2}-4ac}}{2a}=\frac{\mathrm{-}b\pm \sqrt{{b}^{2}-4ac}}{2a}$
The quadratic formula has been derived..
The expression $D={b}^{2}-4ac$ in the root is called the discriminant of the quadratic equation. Since only the root of a positive number or $0$ is itself a real number, it is the value of the discriminant that determines the number of solutions of the equation:
$D>0$ and there are two solutions;
$D=0$ and there is one solution (or the same solution twice);
$D<0$ and there are no real solutions;