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Friday, June 18, 2021
Trigonometry  Trigonometry hand trick  Trigonometry Class 10  Trigonometry tricks  त्रिकोणमिति
Thursday, June 10, 2021
Quadratic Formula  द्विघात समीकरण  Quadratic Equation Tricks  How To Solve Quadratic Equations
Quadratic Formula  द्विघात समीकरण  Quadratic Equation Tricks  How To Solve Quadratic Equations
Youtube Video: Quadratic Formula
An example of a Quadratic Equation:
The function makes nice curves like this one:
Name
The name Quadratic comes from "quad" meaning square, because the variable gets squared (like x^{2}).
It is also called an "Equation of Degree 2" (because of the "2" on the x)
Youtube Video: Quadratic Formula
Standard Form
The Standard Form of a Quadratic Equation looks like this:
 a, b and c are known values. a can't be 0.
 "x" is the variable or unknown (we don't know it yet).
Here are some examples:
2x^{2} + 5x + 3 = 0  In this one a=2, b=5 and c=3  
x^{2} − 3x = 0  This one is a little more tricky:
 
5x − 3 = 0  Oops! This one is not a quadratic equation: it is missing x^{2} (in other words a=0, which means it can't be quadratic) Youtube Video: Quadratic FormulaHidden Quadratic Equations!As we saw before, the Standard Form of a Quadratic Equation is ax^{2} + bx + c = 0 But sometimes a quadratic equation doesn't look like that! For example: 
About the Quadratic Formula
Plus/Minus
First of all what is that plus/minus thing that looks like ± ?
The ± means there are TWO answers:
x = −b + √(b^{2 }− 4ac)2a
x = −b − √(b^{2 }− 4ac)2a
Here is an example with two answers:
But it does not always work out like that!
 Imagine if the curve "just touches" the xaxis.
 Or imagine the curve is so high it doesn't even cross the xaxis!
This is where the "Discriminant" helps us ...
Youtube Video: Quadratic Formula
Discriminant
Do you see b^{2} − 4ac in the formula above? It is called the Discriminant, because it can "discriminate" between the possible types of answer:
Complex solutions? Let's talk about them after we see how to use the formula.
Youtube Video: Quadratic Formula
Using the Quadratic Formula
Just put the values of a, b and c into the Quadratic Formula, and do the calculations.
Example: Solve 5x^{2} + 6x + 1 = 0
Answer: x = −0.2 or x = −1
And we see them on this graph.
Check 0.2:  5×(−0.2)^{2} + 6×(−0.2) + 1 = 5×(0.04) + 6×(−0.2) + 1 = 0.2 − 1.2 + 1 = 0  
Check 1:  5×(−1)^{2} + 6×(−1) + 1 = 5×(1) + 6×(−1) + 1 = 5 − 6 + 1 = 0 
Example: Solve x^{2} − 4x + 6.25 = 0
= −9
√(−9) = 3i
(where i is the imaginary number √−1)
Answer: x = 2 ± 1.5i
The graph does not cross the xaxis. That is why we ended up with complex numbers.
BUT an upsidedown mirror image of our equation does cross the xaxis at 2 ± 1.5 (note: missing the i).
Just an interesting fact for you!
____________________________________________________________
Youtube Video: Quadratic Formula
A quadratic equation is a secondorder polynomial equation in a single variable
(1) 
with . Because it is a secondorder polynomial equation, the fundamental theorem of algebra guarantees that it has two solutions. These solutions may be both real, or both complex.
Among his many other talents, Major General Stanley in Gilbert and Sullivan's operetta the Pirates of Penzance impresses the pirates with his knowledge of quadratic equations in "The Major General's Song" as follows: "I am the very model of a modern MajorGeneral, I've information vegetable, animal, and mineral, I know the kings of England, and I quote the fights historical, From Marathon to Waterloo, in order categorical; I'm very well acquainted too with matters mathematical, I understand equations, both the simple and quadratical, About binomial theorem I'm teeming with a lot o' news With many cheerful facts about the square of the hypotenuse."
The roots can be found by completing the square,
(2) 
src="https://mathworld.wolfram.com/images/equations/QuadraticEquation/NumberedEquation3.gif" style="border: 0px;" width="217" />  (3) 
(4) 
Solving for then gives
(5) 
This equation is known as the quadratic formula.
Here are the steps required to solve a quadratic using the quadratic formula:
Youtube Video: Quadratic Formula
Friday, June 4, 2021
Adding Polynomials  Polynomials Class 10/9  Class 10 Maths Chapter 2  Basics of Polynomial
Polynomial addition and subtraction
Youtube video:
Adding Polynomials  Polynomials Class 10/9  Class 10 Maths Chapter 2  Basics of Polynomial
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maths study 2
Algebraic Expression│Class 7 RS Aggarwal │Introduction of chapter 6
Algebraic Expression│Class 7 RS Aggarwal │Introduction of chapter 6│Class 7 chapter 6 in hindi
What is an Algebraic Expression?
An algebraic expression in mathematics is an expression which is made up of variables and constants, along with algebraic operations (addition, subtraction, etc.). Expressions are made up of terms. They are also termed algebraic equations.
Examples
3x + 4y – 7, 4x – 10, etc.
These expressions are represented with the help of unknown variables, constants and coefficients. The combination of these three (as terms) is said to be an expression. It is to be noted that, unlike the algebraic equation, an algebraic expression has no sides or equal to sign. Some of its examples include
 3x + 2y – 5
 x – 20
 2x2 − 3xy + 5
Variables, Coefficient & Constant
In Algebra we work with Variable, Symbols or Letters whose value is unknown to us.
In the above expression (i.e. 5x – 3),
 x is a variable, whose value is unknown to us which can take any value.
 5 is known as the coefficient of x, as it is a constant value used with the variable term and is well defined.
 3 is the constant value term which has a definite value.
The whole expression is known to be the Binomial term, as it has two unlikely terms.
Types of Algebraic expression
There are 3 main types of algebraic expressions which include:
 Monomial Expression
 Binomial Expression
 Polynomial Expression
Monomial Expression
An algebraic expression which is having only one term is known as a monomial.
Examples of monomial expression include 3x4, 3xy, 3x, 8y, etc.
Binomial Expression
A binomial expression is an algebraic expression which is having two terms, which are unlike.
Examples of binomial include 5xy + 8, xyz + x3, etc.
Polynomial Expression
In general, an expression with more than one terms with nonnegative integral exponents of a variable is known as a polynomial.
Examples of polynomial expression include ax + by + ca, x3 + 2x + 3, etc.
Other Types of Expression:
Apart from monomial, binomial and polynomial types of expressions, an algebraic expression can also be classified into two additional types which are:
 Numeric Expression
 Variable Expression
Numeric Expression
A numeric expression consists of numbers and operations, but never include any variable. Some of the examples of numeric expressions are 10 + 5, 15 ÷ 2, etc.
Variable Expression
A variable expression is an expression which contains variables along with numbers and operation to define an expression. A few examples of a variable expression include 4x + y, 5ab + 33, etc.
Algebraic expression for Class 7
In Class 7, students will come across the terms of algebraic equations such as:
 Coefficient of a term
 Variables
 Constant
 Factors of a term
 Terms of equations
 Like and Unlike terms
Example of using these terms are given below.
If 2x2+3xy+4x+7 is an algebraic expression.
Then, 2x2, 3xy, 4x and 7 are the terms
Coefficient of term x2 = 2
Constant term = 7
Example of like and unlike terms:
Like terms: 2x and 3x
Unlike terms: 2x and 3y
Factors of a term:
If 3xy is a term, then its factors are 3, x and y.
Monomial, Binomial & Trinomial
Also, in grade 7 we will learn about types of expressions, such as monomial, binomial and trinomial. Let us see examples of each.
Monomial: 2x
Binomial: 2x+3y
Trinomial: 2x+3y+9
Addition and Subtraction of Algebraic Expressions
We can add and subtract like terms easily.
Example: Add 3x + 5y – 6z and x – 4y + 2z.
By adding both the expressions we get;
(3x + 5y – 6z) + (x – 4y + 2z)
Separating the like terms and adding them together:
(3x + x) + (5y – 4y) + (6z + 2z)
4x + y – 4z
Formulas
The general algebraic formulas we use to solve the expressions or equations are:
 (a + b)2 = a2 + 2ab + b2
 (a – b)2 = a2 – 2ab + b2
 a2 – b2 = (a – b)(a + b)
 (a + b)3 = a3 + b3 + 3ab(a + b)
 (a – b)3 = a3 – b3 – 3ab(a – b)
 a3 – b3 = (a – b)(a2 + ab + b2)
 a3 + b3 = (a + b)(a2 – ab + b2)
Solved Problem
Example: Simplify the given expressions by combining the like terms and write the type of Algebraic expression. (i) 3xy3 + 9x2 y3 + 5y3x (ii) 7ab2 c2 + 2a3 b2 − 3abc – 5ab2 c2 – 2b2 a3 + 2ab (iii) 50x3 – 20x + 8x + 21x3 – 3x + 15x – 41x3 Solution: Creating a table to find the solution:

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