QUADRATIC POLYNOMIALS AND EQUATIONS (Type 03) x +1/x = m/n
By
Om Jitender Singh Tomar
(ॐ जितेन्द्र सिंह तोमर)
M.A., B.Ed., DNYS, MASSCOM
21/2/21/12/2021
QUADRATIC EQUATIONS
Today, we will discuss with quadratic polynomial, quadratic polynomial are polynomial of the second degree with at least one term.
The general form of a quadratic polynomial is ax² + bx + c.
QUADRATIC EQUATIONS
Now, we will discuss with quadratic equations, quadratic equations are polynomial of the second degree with at least one term.
The general form of a quadratic equations is ax² + bx + c = 0.
QUADRATIC FORMULA
The quadratic formula can be used to solve equations in the standard form. The quadratic formula is usually written as below:
Today we will not discuss, with quadratic equations that are written in the standard form.
We will take quite a bit different quadratic equations which will be converted into the standard form after some effort. We can solve them by using the quadratic formula.
The first special type of quadratic equations we will consider are like the one below:
The first special type of quadratic equations we will consider are like the one below:
1 10
x + __ = ____
3 3
In the traditional method, we would go through the boaring and a hard process of getting the solution of quadratic.
First of all equation is converted into the standard form by using the following steps given below:
(1)
1 10
x + __ = ____
3 3
(1)
1 10
x + __ = ____
3 3
(x² + 1) 10
=> _______ = ____
x 3
=> _______ = ____
x 3
by cross Multiplication
=> 3x² + 3 = 10x becomes
=> 3x² – 10x + 3 = 0
We would then solve it using the quadratic formula by setting
a = 3,
b = –10 and
c = 3.
by quadratic formula
–( b) ± √(b² – 4ac)
= _________________
2a
–( –10) ± √[(–10)² – 4×3×3)
= ___________________________
2 × 3
10 ± √(100 – 36)
= ________________
6
10 ± √(64)
= _________
6
10 ± 8
= ______
6
10 + 8 10–8
x = ______ or x = ________
6 6
x = 18/6 or x = 2/6
x = 3 or x = 1/3
x = 3, 1/3
The solutions are x = 3 and x = 1/3.
Don't afraid, we are not going to solve the equation by this method.
Don't afraid, we are not going to solve the equation by this method.
We are going to solve these by mear observation method.
All we have to do is observe that 10 1 1
__ = 3 ___ = 3 + ___
3 3 3Thus, we would immediately have figured out that we can rewrite the given equation as below:
(1)
1 10
x + __ = ____
3 3
x + __ = ____
3 3
1 1
x + __ = 3 + ____
x 3
x + __ = 3 + ____
x 3
on comparing
=> x = 3, 1/3
or
x + 1/x = 10/3
=> x + 1/x = 3 + 1/3
on comparing
=> x = 3, 1/3
The same method can be used to solve many different problems such as the examples below:
(2)
1 –17
x + __ = ____
x 4
x + __ = ____
x 4
1 1
x + __ = –4 – ___
x 4
x + __ = –4 – ___
x 4
on comparing
=> x = –4, –1/4
or
x + 1/x = –17/4
=> x + 1/x = –4, – 1/4
=> x = –4, –1/4
(3)
1 50
x + __ = ____
3 7
x + __ = ____
3 7
1 1
x + __ = 7 + ____
x 7
x + __ = 7 + ____
x 7
on comparing
=> x = 7, 1/7
or
x + 1/x = 50/7
=> x + 1/x = 7 + 1/7
on comparing
=> x = 7, 1/7
(4)
1 25
x + __ = ____
3 5
x + __ = ____
3 5
1 1
x + __ = 5 + ___
x 5
x + __ = 5 + ___
x 5
on comparing
=> x = 5, 1/5
or
x + 1/x = 26/5
=> x + 1/x = 5 + 1/5
on comparing
=> x = 5, 1/5
We are not restricted to x + 1/x on the left-hand side either. Consider the equation:
(5)
1
1
(2x + 3) + _________ = 50/7
(2x + 3)
Since the right-hand side of the equation can be expanded to 7 + 1/7, by the symmetry of the equation, we can equate 2x + 3 to either 7 or 1/7.
1 1
(2x + 3) + _________ = 7 + ___
(2x + 3) 7
(2x + 3) = 7 or (2x + 3) = 1/ 7
2x + 3 = 7 or 14x + 21 = 1
2x = 7 –3 or 14x = 1 –21
2x = 4 or 14x = –20
x = 4/2 or x = –20/14
x = 2 or x = –10/7
We will get the solutions x = 2 and x = -10/7 to the given equation.
Similarly, consider the equation:
Similarly, consider the equation:
5x (2x + 3) 26
________ + _________ = ____
(2x + 3) 5x 5
5x (2x + 3) 1
________ + _________ = 5 + ____
(2x + 3) 5x 5
The right-hand side of the equation can be expanded to 5 + 1/5. Thus, we can use the symmetry of the resulting equation to derive the following linear equations:
5x 5x 1
________ = 5 or _______ = ___
(2x + 3) (2x + 3) 5
5x 5 5x 1
________ = ____ or _______ = ___
(2x + 3) 1 (2x + 3) 5
5(2x + 3) = 5x or 5(5x) = (2x+3)
10x + 15 = 5x or 25x = 2x+3 10x – 5x = 15 or 25x –2x = 3
5x = 15 or 23x = 3
x = –15/5 or x = 3/23
x = –3 or x = 3/23
5x/(2x + 3) = 1/5 or alternatively, (2x + 3)/5x = 5
These two equations can then be solved to give us x = –3 or x = 23/3.
Now, consider the equation:
(x + 3)/(3x + 5) + (3x + 5)/(x + 3) = 17/4
The right-hand side of the equation can be expanded to 4 + 1/4. This then lets us solve the equation by deriving the linear equations below:
(x + 3)/(3x + 5) = 4
(x + 3)/(3x + 5) = 1/4 or alternatively, (3x + 5)/(x + 3) = 4
These equations can then be solved to give us x = -7 and x = -17/11.
Now, consider an equation of the type below:
x - 1/x = 3/2
We can rewrite the equation as below:
x - 1/x = 2 - 1/2
We may be tempted to conclude from the symmetry of the equation on both sides of the equal-to sign that x = 2 or x = 1/2. That would be wrong. In equations such as the above where the terms are connected by "-" signs instead of "+" signs, the solutions are x = 2 and x = -1/2. Only with x = -1/2 is it possible to get -1/x = 2, and therefore x - 1/x = 2 - 1/2. This is important to remember.
We will illustrate this with a few examples as below:
x - 1/x = 8/3 becomes
x - 1/x = 3 - 1/3, which then leads to x = 3 and x = -1/3 as the solutions.
x - 1/x = 63/8 becomes
x - 1/x = 8 - 1/8, which then leads to x = 8 and x = -1/8 as the solutions.
x - 1/x = -24/5 becomes
x - 1/x = -5 + 1/5 which then leads to x = -5 and x = 1/5 as the solutions.
The technique is equally applicable to cases where the left-hand side consists of other terms than x and 1/x. The following examples illustrate a few examples of these cases:
(3x + 2) – 1/(3x + 2) = 63/8
These two equations can then be solved to give us x = –3 or x = 23/3.
Now, consider the equation:
(x + 3)/(3x + 5) + (3x + 5)/(x + 3) = 17/4
The right-hand side of the equation can be expanded to 4 + 1/4. This then lets us solve the equation by deriving the linear equations below:
(x + 3)/(3x + 5) = 4
(x + 3)/(3x + 5) = 1/4 or alternatively, (3x + 5)/(x + 3) = 4
These equations can then be solved to give us x = -7 and x = -17/11.
Now, consider an equation of the type below:
x - 1/x = 3/2
We can rewrite the equation as below:
x - 1/x = 2 - 1/2
We may be tempted to conclude from the symmetry of the equation on both sides of the equal-to sign that x = 2 or x = 1/2. That would be wrong. In equations such as the above where the terms are connected by "-" signs instead of "+" signs, the solutions are x = 2 and x = -1/2. Only with x = -1/2 is it possible to get -1/x = 2, and therefore x - 1/x = 2 - 1/2. This is important to remember.
We will illustrate this with a few examples as below:
x - 1/x = 8/3 becomes
x - 1/x = 3 - 1/3, which then leads to x = 3 and x = -1/3 as the solutions.
x - 1/x = 63/8 becomes
x - 1/x = 8 - 1/8, which then leads to x = 8 and x = -1/8 as the solutions.
x - 1/x = -24/5 becomes
x - 1/x = -5 + 1/5 which then leads to x = -5 and x = 1/5 as the solutions.
The technique is equally applicable to cases where the left-hand side consists of other terms than x and 1/x. The following examples illustrate a few examples of these cases:
(3x + 2) – 1/(3x + 2) = 63/8
=> (3x + 2) – 1/(3x + 2) = 8 – 1/8
=> 3x + 2 = 8, 3x + 2 = –1/8
=> x = 2, x = –17/24
2x/(5x + 1) – (5x + 1)/2x = –15/4
=> 2x/(5x + 1) – (5x + 1)/2x = 1/4 – 4 => 2x/(5x+1) = 1/4, 2x/(5x + 1) = –4
=> x = 1/3, –2/11
(4x + 3)/(3x + 4) – (3x + 4)/(4x + 3) = 24/5
=> (4x + 3)/(3x + 4) – (3x + 4)/(4x + 3) = 5 – 1/5
=>(4x + 3)/(3x + 4) = 5, (4x + 3)/(3x + 4) = –1/5
=> x = –17/11, –19/23
Sometimes, the equation may have undergone some transformations that hide its true nature. Consider the equation below:
x + (x + 1)/x = 7/2
The equation can actually be simplified as below:
x + (x + 1)/x = 7/2 becomes
x + x/x + 1/x = 7/2 becomes
x + 1/x = 7/2 – 1 becomes
x + 1/x = 5/2 becomes
x + 1/x = 2 + 1/2 which directly leads to the solutions x = 2 and x = 1/2.
Many of these transformations are difficult to peal back on sight to reveal the true nature of the equation. But it may be worthwhile to examine the equation and try a few transformations to see if any simplifications are possible before giving up and solving it using the traditional method.
Sometimes, the right-hand side is not as easily decomposed to a pair of reciprocals as in the above examples. Consider the equation below:
x + 1/x = 25/12
The right-hand side may look as if it can not be converted into a pair of reciprocals since at first glance, it breaks apart as 2 and 1/12. But a closer look will reveal that 25/12 = 3/4 + 4/3. Thus, we can actually rewrite the equation above as:
x + 1/x = 3/4 + 4/3
This then leads immediately to the solutions x = 3/4 and x = 4/3 by the symmetry of the equation on both sides of the equal-to sign.
Let us tackle a few more equations like the above:
x + 1/x = 13/6
Sometimes, the equation may have undergone some transformations that hide its true nature. Consider the equation below:
x + (x + 1)/x = 7/2
The equation can actually be simplified as below:
x + (x + 1)/x = 7/2 becomes
x + x/x + 1/x = 7/2 becomes
x + 1/x = 7/2 – 1 becomes
x + 1/x = 5/2 becomes
x + 1/x = 2 + 1/2 which directly leads to the solutions x = 2 and x = 1/2.
Many of these transformations are difficult to peal back on sight to reveal the true nature of the equation. But it may be worthwhile to examine the equation and try a few transformations to see if any simplifications are possible before giving up and solving it using the traditional method.
Sometimes, the right-hand side is not as easily decomposed to a pair of reciprocals as in the above examples. Consider the equation below:
x + 1/x = 25/12
The right-hand side may look as if it can not be converted into a pair of reciprocals since at first glance, it breaks apart as 2 and 1/12. But a closer look will reveal that 25/12 = 3/4 + 4/3. Thus, we can actually rewrite the equation above as:
x + 1/x = 3/4 + 4/3
This then leads immediately to the solutions x = 3/4 and x = 4/3 by the symmetry of the equation on both sides of the equal-to sign.
Let us tackle a few more equations like the above:
x + 1/x = 13/6
=> x + 1/x = 2/3 + 3/2
=> x = 2/3, 3/2
x + 1/x = 29/10
=> x + 1/x = 2/5 + 5/2
=> x = 2/5, 5/2
Similarly, we can solve the equations below also by recognizing the right-hand side to be a difference of reciprocals:
x – 1/x = 11/30
Similarly, we can solve the equations below also by recognizing the right-hand side to be a difference of reciprocals:
x – 1/x = 11/30
=> x – 1/x = 6/5 – 5/6
=> x = 6/5, –5/6
x – 1/x = –40/21
=> x – 1/x = 3/7 - 7/3
=> x = 3/7, –7/3
This same technique can be extended to equations where the left-hand side is a sum or difference of any two reciprocal quantities, not just x and 1/x. I will leave those extensions to the reader in the interest of keeping this lesson from growing any longer than it already has become!
Assuming that the left-hand side is a pair of reciprocals, connected by either "+" or "-", how do we verify whether the right-hand side can be expressed as a pair of reciprocals with the same sign between them? We will explore this question in greater detail in the next lesson.
In the meantime, I hope you have found this lesson useful and interesting. I also hope you will apply the techniques explained in this lesson on real problems so that you become familiar not only with the technique itself (which is actually quite trivial), but also with the fractions that result either from the addition of numbers with their reciprocals, or the differences between numbers and their reciprocals. That will enable one to apply this technique where appropriate, on sight and mentally, to solve the types of quadratic equations we have dealt with in this lesson. Good luck, and happy computing!
This same technique can be extended to equations where the left-hand side is a sum or difference of any two reciprocal quantities, not just x and 1/x. I will leave those extensions to the reader in the interest of keeping this lesson from growing any longer than it already has become!
Assuming that the left-hand side is a pair of reciprocals, connected by either "+" or "-", how do we verify whether the right-hand side can be expressed as a pair of reciprocals with the same sign between them? We will explore this question in greater detail in the next lesson.
In the meantime, I hope you have found this lesson useful and interesting. I also hope you will apply the techniques explained in this lesson on real problems so that you become familiar not only with the technique itself (which is actually quite trivial), but also with the fractions that result either from the addition of numbers with their reciprocals, or the differences between numbers and their reciprocals. That will enable one to apply this technique where appropriate, on sight and mentally, to solve the types of quadratic equations we have dealt with in this lesson. Good luck, and happy computing!
Post a Comment
Post a Comment