An inclined plane problem solution
Quadratic Equations
Important Information
(on-the-table
material for everybody)
1. For any real
numbers a and b the identity
a2 + 2 ab
+ b2 = (a + b)2
is valid.
Hence: for any numbers
p and q the identity (the pq- formula)
p2 + pq =
(p + q/2)2 - q2/4
is valid
2. If A2 =
B and B > 0 hence, A = √B and A = -√B (the ÀÂ - formula).
If B is a negative
number, hence real number A dose not exists.
If B = 0, hence A = 0
3. Indicators of a
Quadratic Equation.
A quadratic equation
may content any amount of terms.
a) Terms/addends may
be
- numbers;
- multiplications of a
number by the first or the second power of an unknown quantity.
b) A quadratic
equation must contents always the second power of an unknown
quantity.
c) There must not present any other terms except
of a and b.
4. The Algorithm for
Solving of a Quadratic Equation by Completing the Square
a) Separate the terms
with an unknown from numbers, collecting all the numbers on the right side of
the equation, and all the other terms on the left side.
b) Simplify the expression,
summarize all the terms, which contents the same power of an unknown; rearrange
all the terms placing the second power term of an unknown on the first place
and the number on the last one.
After this step, we
should get the equation like the following:
ax2
+ bx = d
c) Divide both sides
of the equation by a number factor a
of the second power term.
d) Apply the
pq-formula to the left side of the equation to complete the square (treat x as
p and b/a as q).
e) Add to both sides
of the equation the q2/4 term.
f) Apply the ÀÂ-formula.
g) Find an unknown
(solve two simple linier equations).
5. The Algorithm for
Solving of a Quadratic Equation
a) Convert a quadratic
equation into a general form.
b) Determine the
coefficients and the free term of the equation.
c) Calculate the
discriminant.
d) Apply the quadratic
formula to find the roots of the equation.
The lecture
Imagine that I am your
teacher, and you are my students.
Let’ start.
Please, any question
about the homework.
There are no questions.
Perfect. Now I am really sure that the last lesson was good!
This page contains the
rules, properties and other important information, which we shall apply today.
Nevertheless, let me to
test a little bit what you remember form the information which will be
necessary for us today.
On the page, there is
a set of the equations. Some of them are quadratic equations, some other are
not. Now I am going to call the equation, and you have to raise up the right
hand, if this equation is a quadratic equation, and just sit still if it is not
(why not quadratic).
Ok, now we have a set
of quadratic equations. We may solve any of them with the help of our
algorithm. You can recall the algorithm if take a look on the page. Now I want
to present you very important problem. Dose the algorithm depends on numbers,
which are used in a quadratic equation? No. We may write any numbers in a
quadratic equation, the algorithm for its solving will be the same. But if a
property, a rule, an algorithm are valid for any numbers, it means we may
replace these numbers by letters with no breaking validity of property, a rule,
an algorithm. If instead of these numbers 2 and 8 I shall write letters R and
S, can I apply the algorithm to solve the quadratic equations? Certainly! I do
not see any number over here, what dose it mean? It means, that number 1 is
written here, but it is written by invisible ink. Can I replace it number by a
letter? Certainly. Will the algorithm be
different after this? No.
Therefore, we may take
any concrete quadratic equation and replace its number by letters, and then we
can apply our algorithm for solving quadratic equations and find the roots.
After this we may use the obtained formulas for solving any concrete quadratic
equation. Let’s begin to perform this program.
The first problem,
which we meet, is what kind of an equation should be taken as a starting
equation? We have a set of concrete quadratic equations. Let's take one of them
and apply the algorithm.
The first step is
separating the terms. We want to move a number 5 on the left side, and the
terms 6x and -2x2 on the right side. For this purpose we should add
to both parts of the equation terms -5 - 6x + 2x2. Let's make it.
Now we shall simplify the expression. Let’s write aside each other terms with
the same power of an unknown. We have two terms, which have the second power of
an unknown. We may add them. (By the way, what the property of real numbers we
have just applied? The answer is the distributive property). Do the same for
this pair of terms (also, we have applied the commutative property).
In the result, we have
the equation such …
We saw on the last
lesson, that after the step b) all quadratic equations have an equally look. It
means it is convenient substituting all the numbers by letters now. Let's do
it. Say me letters that you want to use. Take * instead of -8, ** instead of 2
(what 2? This or this?). And we can write *** instead of this 2. At last
replace 1 by letter ****. Now let’s try to apply the algorithm to solve this
equation.
The next step is
dividing both sides of the equation by the factor of the second power term.
Stop. Where is the second power of an unknown? It has disappeared. What wrong
was done? We have replaced this 2 by the letter. It was our mistake. We may use
letters instead of numbers, but we may not change a type of an equation. The
equation was quadratic and should remain quadratic. Why? Because our algorithm
is applicable for quadratic equations only!
We have to return a
power at a former place. OK. Now we have the following equation
ax2
+ bx + c = 0
So, we have a
quadratic equation. There is just a single number in this equation, 2, which is
saying that this equation is quadratic.
Now we can apply our
algorithm to solve it. The next step of the algorithm is
c) divide both sides
of the equation by a number factor of the second power term.
I.e. we should
multiply both parts by reciprocal of this factor.
d) Apply the
pq-formula to complete the square.
Instead of p and q
letters, we have these letters.
However, we know the
main property of algebra.
The result of
calculations does not depend on letters, which we use for our variables or
quantities.
It means, instead of p
we may write *, and instead of q **.
Hence, we receive the
following expression.
The next step is e)
e) Add to both sides
of the equation the q2/4 term. OK.
These two terms are
canceled out.
At last
f) Apply the ÀÂ-formula.
First, magnitude of
the right side of the expression should be positive (or zero).
If it is not, hence
the equation has no any real roots.
If it is, we have the
solution in real numbers.
Mathematicians do not
love expression where there is an addition of fractions. Therefore, we will
simplify this expression. These fractions have the least common denominator,
which is equal to ***.
We may multiply
numerator and a denominator of the first fraction on **. Now we can add the
fractions. It leads to such an expression.
Now we apply ÀÂ - the formula.
It gives to us these expressions.
Let’s make the last
step of the algorithm.
g) Find the roots of
the quadratic equation.
We simply should add
*** to both sides of equality.
In the end, we obtain
the answer.
x =
x =
I have to say, that
nobody in the whole world uses these letters to write and to solve a quadratic
equation. We certainly may use them, but other mathematicians will think that
we are too original.
You can find in any
textbook, the common view of a quadratic equation is
ax2
+ bx + c = 0
This expression is
called the general quadratic equation or a quadratic equation in the general
form.
Numbers a and b are
called coefficients of the equation. The last number c (which is do not touch
any unknown) is called a free term of the equation.
We easy can transform
the general quadratic equation into ours. We have simply to move the c number
from the left side to the right side. Now instead of * we may write a, instead
of ** - b, and instead of *** we should write -c
If you make up some
algebra, you will receive the following result.
x =
x =
This result is called
the quadratic formula.
The expression = D is called the
discriminant of the quadratic equation. The letter D is a standard denotation
for the discriminant.
If the discriminant is
negative, the quadratic equation has no any real roots.
OK, let’s finish for
today making the final conclusions.
If you have any
quadratic equation you can always rewrite it in the general form. After this,
you can apply the quadratic formula to find the roots.
The algorithm for
solving of a quadratic equation consists now just of four steps.
à) Rewrite a quadratic equation in a
general form.
b) Identify
coefficients and a free term of the equation.
c) Calculate the
discriminant.
d) Apply the quadratic
formula.
In the following
lesson, we shall train solving quadratic equations by the quadratic formula. I
want to give a small homework to you.
1. Derive the
quadratic formula.
2. With the help of
direct substitution in the equation, prove that these expressions really are
roots of this equation.
Questions?
Thank you for your
work.
The training
In the previous
lesson, we have derived the quadratic formula, which is the general solution of
a general quadratic equation.
The algorithm for
using the quadratic formula is very simple.
The algorithm for
solving of a quadratic equation now consists of four items.
à) Rewrite a quadratic equation in a
general form.
b) Identify
coefficients and a free term of the equation.
c) Calculate the
discriminant.
d) Apply the quadratic
formula.
Let's train solving
quadratic equations by applying the new algorithm. We will take the equations,
which we have already solved.
Who will take a risk
to go to the blackboard?
We should hold in front
of our eyes a general quadratic equation and compare our concrete equation with
it. The first question, which will help us to apply our algorithm, is what
difference is there between the concrete equation and the general equation?
The first difference,
which we can see, is an order of terms in the equations.
The general equation
has always a term with x2 on the first place, then goes term with õ, and there is just a number on the last place.
The first step of our solution
is moving all the terms no the right side of the equation, leaving in the
equation three terms only, and to rearrange them in the right order. Let's make
it.
Perfect.
Now we should perform
the second step of the algorithm.
For this purpose let's
write both equations (the general one and the certain one) exactly one under
the second.
Take a close look at
the equations and again ask yourself, what is the difference between the
equations?
Or, I can formulate
this question in other words. What numbers have to be written instead of
letters a b and c to obtain the certain equation from the general one?
The letter a should be
equal *, b should be equal **, c should be equal ***.
All, what we have to
make now, it is just substituting the obtained values of coefficients and a
free term into the expressions for the discriminant and the roots.
Let's do it.
Perfect.
The equation is
solved. How can we check ourselves and prove the result? Yes, we can plug the
numbers into the initial equation. After doing some arithmetic, we should get
0.
Thank you.
Who would like to
solve the next equation?
Now, let's solve some
non-quadratic equations, which, however, may be converted to the general
quadratic equation.
Homework.