EQUATIONS
Welcome to the world of algebra. Do you know what an equation is? An equation simply has an equal-to sign and some numbers or variables operated by some signs. Example can be 2 + 3 = 5. But this is a very basic example which we all have been learning since our childhood. Equations basically have LHS (left hand side) and RHS (right hand side) and in between them is an equal-to sign which tells us that LHS is always equal to the RHS. Let's take another example to make things more clear.
Say, x + 4 = 7
Here we know that x is a variable. Now what is a variable? Well, a variable is something that can take any value. It is not fixed. And their contrary are constants which have a fixed value. All the numbers we see are constants like 1,2, 100, 586, -4, √5 etc, whereas we generally represent variables with alphabets or symbols such as x, y, α, β etc. You can use any alphabet or symbol to represent a variable. Did you know that the most popular alphabet used as a variable is x?
LINEAR EQUATIONS
Now, equations with only one variable are known as linear equations in one variable. Now what is a linear equation in one variable? Well, linear means that the highest power of the variable in the equation is 1. In the example x + 4 = 7, the variable x's power is 1 and x is the only variable in the equation, hence the equation x + 4 = 7 is a linear equation in one variable. Note that the highest power of a variable is known as the degree of the equation. In this case, the degree of the equation is 1.
We have what's called a general form of a linear equation. We generalize things in such a way that it holds true for any case in that concept. The general form of a linear equation is:
ax + b = 0, where a ≠ 0, a and b are real numbers.
Here x is the variable with power 1. What are 'a' and 'b' here? Well, they are constants. a is the constant coefficient of the variable x and b is a separate constant value. Now why is a ≠ 0? Well if we put a = 0 then,
0x + b = 0
⇒ b = 0
This is clearly not a linear equation and moreover this will lead the value of b to always be equal to zero which is not true. Hence a can not have the value zero.
In the general form, it is also mentioned that a and b are real numbers. Now what are real numbers? Are there any unreal numbers? To get the answers and know more about numbers visit this article: Concept Of Infinity. Basically real numbers are just a collection of rational and irrational numbers.
Let's take a final example to understand.
3x - 6 = 9
⇒ 3x - 6 - 9 = 0
⇒ 3x - 15 = 0
Here a = 3, b = -15
In the later part, we will be looking how to solve such equations as of course, variable also have some value but it keeps changing as we change the equations.
QUADRATIC EQUATIONS
There can be more variables and higher degrees than just one, of course. We know that when the degree of an equation is 1, then the equation is a linear equation. Then what do we call an equation with degree 2 i.e. when the highest power of the variable is 2? Well, it is called a "quadratic equation". ‘Quad’ means four, but ‘Quadratic’ means ‘to make square’. Similarly, degree 3 is a cubic equation and degree 4 is a biquadratic equation.
A good example of a quadratic equation will be:
x² + x + 1 = 0
Here as you can see, the degree or the highest power of the variable x is 2. Let's have a look at the general form of a quadratic equation:
ax² + bx + c = 0, a ≠ 0; a, b and c are real numbers
The same thing follows for quadratic equations as well. a, b and c are constants, or rather, real constants. Here also, if a = 0, then it will no longer be a quadratic equation. It will then become a linear equation. Now why is there no b ≠ 0 mentioned in the general form. Well it is not necessary for b to not be zero. b can be zero because it does not affect the degree of the equation. It will remain quadratic even if the value of b is zero.
Let's take an example:
3x² + 4x = 3
⇒ 3x² + 4x - 3 = 0
Here a = 3, b = 4 and c = -3
Okay, now we know some basic ideas about what an equation really looks like. Now of course as I mentioned earlier there can be more equations with higher powers such as 3, 4 and so on with more than one variables such as y, z, a and so on. Those equations also follow the same concept as we saw in the case of quadratic and linear equations. The only thing which changes is the way of solving or finding the value(s) of the variable(s). Here we will look at the procedure to solving linear and quadratic equations as these are more commonly used in our real life activities. Let's start with solving linear equations in one variable.
SOLVING LINEAR EQUATIONS
Linear equations, as we've seen, have degree 1 and hence are easier to solve when compared to quadratic equations. Linear equations in one variable can be solved simply by simplification. Let's take an example.
x - 5 = 10
Simplification of the equation gives us:
⇒ x = 10 + 5
⇒ x = 15
Hence we got the value of x by just simplification. We can also add more x's as ultimately the degree is still going to remain 1 and it will still be a linear equation in one variable until we use only x's and not any other variables such as y's. Let's take another example:
x + 14 = 4x - 8
Now we should know that we can actually add the similar variables. In this case we have x and 4x which are also called like terms as they have the same variable. They can be added, subtracted, multiplied and divided as per the equation. Now let's solve it.
Simplification of the equation gives us:
x + 13 = 4x - 8
⇒ x - 4x = - 8 - 13
⇒ -3x = -21
⇒ 3x = 21
⇒ x = 7
With these examples, we see that solving linear equations do not require any special procedure. They can be solved simply by the process of simplification. Let us now solve quadratic equations which do have a lot of special methods for simplification.
SOLVING QUADRATIC EQUATIONS
Now, quadratic equations can be solved by two different methods, namely:
1) Factorization Method
2) Quadratic Formula
(Shreedharacharya's Method)
FACTORIZATION
Now what do you mean by factorization method? Well we know what factors are, right? A factor when multiplied with another particular factor gives a number which also their multiple. Example: 6*2 = 12. Hence, the factors of 12 are 6 and 2 and 12 is the multiple of 6 and 2. Now of course 12 can be calculated by multiplying other numbers as well which all are its factors as well. So what we basically mean by factorization is, we are trying to split the given equation into its factors which help us find the values of the variables. Note that a quadratic equation can have a maximum of two solutions since its degree is 2.
Now let us take an example to understand factorization.
Solve for x:
x² + 2x + 1 = 0
⇒ x² + x + x + 1 = 0.....[Splitting 2x into x + x]
⇒ x(x + 1) + 1(x + 1) = 0.....[Taking common out]
⇒ (x + 1) (x + 1) = 0
Here either of the factors must be equal to 0. Since both the factors are the same, hence we proceed:
x + 1 = 0
⇒ x = -1
Therefore, the value of x in the quadratic equations -1. Here we get only one solution. By the way, solutions of a quadratic equation are known as "roots".
QUADRATIC FORMULA
The quadratic formula is another way of finding the roots. Given by the ancient Indian mathematician Shreedharacharya in 1025 AD, it is also known as the Shreedharacharya's method. Now what is the quadratic formula? Well the roots for x as per the quadratic formula can be shown like this: [x = - b ± √(b² - 4ac) / 2a]
Now this might look absurd to you at first but if you understand where this formula came from, then you will easily understand what this means. We will give a proof for the quadratic formula in our next article. Quadratic formula is useful in solving complex quadratic equations that can not be easily factorized.
x = - b ± √(b² - 4ac) / 2a
In this formula, the term b² - 4ac is also known as the discriminant (D) which tells us the nature of roots. As you can see the discriminant has a plus-minus (±) sign which implies that it can be both positive and negative since it is a square root. Square roots can be both positive and negative. Hence we separately operate plus and minus and hence get two values of x which are the roots of the quadratic equation. Let's see an example.
Solve for x:
x² + 6x + 5 = 0
Using the quadratic formula:
x = - b ± √(b² - 4ac) / 2a
x = - (6) ± √[(6)² - 4(1)(5)] / 2(1)
x = - 6 ± √(36 - 20) / 2
x = - 6 ± √16 / 2
x = - 6 ± 4 / 2
Case 1: x = - 6 + 4 / 2
x = - 2 / 2
x = -1
Case 2: x = - 6 - 4 / 2
x = - 10 / 2
x = -5
Therefore the roots are -1 and -5.
We are now done with the method of solving quadratic equations. Here, some of you might say another method of finding the roots: completing the square method. Well, yes, it also a method of solving a quadratic equation. But this method is just a primitive version of the quadratic formula. What do I mean by this? Well, we will understand how completing the square method helps us get the quadratic formula in the next article. So completing the square is basically the procedure of making a whole square so that there is a whole square formed.
So that was all in today's article. I hope you learnt all about equations and how to solve them.
Post a Comment