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Δβα\Delta _{\beta \alpha}
01
Linear equations
Δβα\Delta _{\beta \alpha}
02
Quadratic equations
Δβα\Delta _{\beta \alpha}
02
Quadratic equations
Starter
List out everything you know about quadratic equations. This can be things like the formula, or if you already know the concept of a discriminant, that too!
Introduction to the discriminant
You should already know the different forms you can express a quadratic in which can provide different types of solutions (i.e.
f(x)=ax2+bx+cf(x)=ax^2+bx+c
for roots,
f(x)=a(x−r1)(x−r2)f(x)=a(x-r_1)(x-r_2)
again for roots, and
f(x)=a(x−h)2+k)f(x)=a(x-h)^2+k)
for vertex points), so I won't bother explaining each, and instead we'll jump straight to the interesting bits: roots. Roots are the
xx
-intercepts of a polynomial function (such as a quadratic). Within a quadratic, you can have either 2, 1, or 0 roots depending on the nature of the function. We can find the number of these roots in a given quadratic through what's called the discriminant:
b2−4acb^2-4ac
. The note below describes the solutions and their cases.
Note N02.0a - Solution occurrences
b2−4ac<0b^2-4ac<0
when the quadratic has
00
real solutions
b2−4ac=0b^2-4ac=0
when the quadratic has
11
real solution
b2−4ac>0b^2-4ac>0
when the quadratic has
22
distinct solutions
Think about it, when you get a math error on your calculator when typing a quadratic into the formula
−b±b2−4ac2a\frac{-b \pm \sqrt{b^2-4ac}}{2a}
, chances are, the discriminant (the
b2−4acb^2-4ac
bit), is
<0<0
. You'll learn what happens when you square-root a negative number, in a later chapter. For now, just assume that you get no real solution when the discriminant is less than 0.
Using the
aa
coefficient and discriminant
When sketching a quadratic, we can use the discriminant to determine how many intercepts we have, but we can also use the
aa
coefficient (that's negative or positive) to determine the "directionality" that the quadratic points. If the coefficient of that first
xx
is negative, the graph points downwards i.e.
−x2+2x+1-x^2+2x+1
(go ahead and plot this for yourself). If that first variable is positive, the graph points upwards i.e.
x2+2x+1x^2+2x+1
. This should be common knowledge, hence this section is short.