Quadratic Equations (Level 1) In the following series of videos we will
cover the various methods and techniques required to solve quadratic equations. Recall that
a quadratic equation is an equation of the form: ax squared plus bx plus c equals 0,
where a and b are coefficients, and c is a constant. In addition a cannot equal to 0.
A quadratic equation is a specific type of polynomial it is also known as a polynomial
equation of degree 2 or a second degree polynomial equation. Lastly, it is also known as a second
order polynomial equation. All of these names describe the same type of polynomial. Notice
that a quadratic equation tends to have 3 terms the term ax squared is the quadratic
term or second degree term. This term is the reason why this polynomial is quadratic in
form this is why a cannot equal 0 because if a equal 0 this term will disappear and
you are left with a polynomial that is not quadratic in form, notice that this term contains
a variable that is raised to the power of 2, also notice that this is the term
that contains the highest power of any of the three remaining terms, this is a tell tell
sign that you are dealing with a polynomial that is quadratic in form, in the same manner
the term bx is the linear term or first degree term because the power on the variable x is
equal to 1 even though it’s not explicitly shown the variable is implied to be raised
to the power of 1, and finally the term c is the constant term or the 0 degree term because
the power of the variable is equal to zero and this results in x raised to the power of 0 which
is equal to 1 which is then multiplied by the constant and simplifies to the constant
c. Quadratic equations do not always contain 3
terms, for the most part there are 3 types of quadratic equations they include quadratic
monomials such as, 2x squared equals 0, quadratic binomials such as negative x squared minus
4 equals 0, and negative 5x squared plus 2x equals 0 and quadratic trinomials such as
3x squared plus 2x plus 4 equals 0, notice that a common characteristic of each quadratic
equation is the fact that there exist a term that contains a variable raised to the power
of 2 and it’s also the highest power that is found out of all the terms. Remember this
is the defining characteristic of a quadratic equation; specifically the polynomial must
contain a power function of degree 2 in other words an x squared term, and this power must
be the highest out of the remaining terms. Now let’s talk about the standard form of
a quadratic equation. When solving quadratic equations we tend to
organize the terms in an order suitable for solving, we usually want to organize or rewrite
the polynomial in descending order and equal to zero. For example the following quadratic
trinomial is not in standard form, four minus 4x plus x squared equals 0.
From left to right the degree or order of each term
is 0, 1, and 2 respectively, we want it in descending order so we need to rewrite and
organize it as follows, x squared minus 4x plus 4 equals 0.
Notice that the degree is now written in descending order from left
to right its 2, 1, and 0. This is the form you want to organize all your quadratic equations
before you start solving them. In the same manner the following quadratic binomials are
not in standard form, 5 plus 5x sqaured equals 0 and 9x minus 2x squared equals 0.
The degrees of the first quadratic equation are, from left to the right
0 and 2 respectively, so we want to organize them with the degrees arranged from 2 to 0
respectively as follows, 5x squared plus 5 equals 0.
Likewise the degree on the second quadratic binomial are arranged
from 1 to 2 respectively, so we want the terms organized with degrees arranged from 2 to
1 respectively, as follows, negative 2x squared plus 9x equals 0,
and finally for quadratic monomials there is no need to organize
it since it only contains a term of degree two.
Such as 2x squared equals 0.
Alright, now let’s talk about what it means to solve for a quadratic equation, recall that when
you solved an equation such as the linear equation 2x plus 4 equals 0 also known as
a polynomial equation of degree 1 because the highest degree of the polynomial is 1
and linear because when you graph the left hand side of the equation you generate a line.
Let’s go ahead and actually graph it. Noticing that the left hand side of the equation is
nothing more than a line of the form y = mx + b where m is the slope and b is the y intercept
in this case our slope is equal to 2 and our y intercept is equal to positive 4.
When you are solving any equation in algebra you are actually trying to figure out where
a specific function in this case the line 2x + 4 is equal to zero in other words where
does it cross the x-axis? So looking at our graph it seems that this occurs when x = -2,
So let’s actually solve this equation using algebraic techniques as oppose to graphical
techniques. So we subtract 4 from both sides as follows, then we divide by 2 on both sides
to solve for x which ends up being equal to negative 2. This represents the point in which
the graph of the line intersects the x-axis specifically at (-2, 0).
Now let’s actually talk about quadratic equations, when we graph quadratic equations
they usually have the following form, this U shape form is called a parabola, and just
like lines which contain positive and negative slopes parabolas can open upward or downwards.
We will learn how to graph parabolas in a later video for now I want you to understand
how quadratic functions look like when you graph them.
Whenever you are solving quadratic equations or any equation in algebra you are always
looking for the value of x that will make the equation equal to 0, in other words you
are looking for the x intercept of the function when graphed. Now this is where the similarities
between lines and quadratics end, when we graph lines they always intersect the x-axis
once hence linear equations have one solution, but when dealing with quadratics you always
have two solutions, the graph of quadratic polynomials can look like this where the parabola can
cross the x-axis twice, hence you have 2 real solutions, or it can touch the axis once like
this, in this case you still have two solutions this is what is called a repeated solution more about
this later, and lastly the graph of a quadratic can look like this where it does not cross
the x-axis whatsoever, in this case you have no real solutions but you do have complex
or imaginary solutions. In either case you always have 2 solutions when solving quadratic
equations. So make sure you keep this detail in mind when we start solving quadratic equations.
Alright in our next video we will start solving quadratic monomials and binomials.