An inclined plane problem solution

The Lesson 1 example

The fundamentals of Physics

 

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.