Grade 8
  1. Number Systems  1.1. Rational Numbers  1.2. Irrational Numbers  1.3. Real Numbers  1.4. Properties of Real Numbers  1.4.1. Commutative Property  1.4.2. Associative Property  1.4.3. Distributive Property  1.5. Representation on the Number Line  1.6. Operations on Real Numbers  1.6.1. Addition and Subtraction  1.6.2. Multiplication and Division  1.7. Rationalization  1.8. Exponents and Powers  1.9. Square Roots and Cube Roots
  2. Algebra  2.1. Algebraic Expressions  2.2. Identities and Simplification  2.2.1. Multiplying Binomials  2.2.2. Using Algebraic Identities  2.3. Factorization  2.3.1. Common Factors  2.3.2. Factorization by Grouping  2.3.3. Factorization of Trinomials  2.3.4. Special Products  2.4. Linear Equations in One Variable  2.4.1. Solving Linear Equations  2.4.2. Word Problems on Linear Equations
  3. Geometry  3.1. Understanding Quadrilaterals  3.1.1. Properties of Quadrilaterals  3.1.2. Types of Quadrilaterals  3.1.2.1. Parallelogram  3.1.2.2. Rectangle  3.1.2.3. Square  3.1.2.4. Rhombus  3.1.2.5. Trapezium  3.2. Constructions  3.2.1. Constructing Quadrilaterals  3.2.2. Constructing Bisectors  3.2.3. Constructing Angles  3.2.4. Constructing Triangles  3.3. Symmetry and Transformations  3.3.1. Reflection  3.3.2. Rotation  3.3.3. Translation  3.3.4. Symmetry in Patterns
  4. Mensuration  4.1. Area and Perimeter  4.1.1. Perimeter of Polygons  4.1.2. Area of Triangles  4.1.3. Area of Quadrilaterals  4.1.4. Area of Circles  4.2. Surface Area and Volume  4.2.1. Cubes  4.2.2. Cuboids  4.2.3. Cylinders  4.2.4. Cones  4.2.5. Spheres
  5. Data Handling  5.1. Organizing Data  5.2. Frequency Distribution Tables  5.3. Graphical Representation of Data  5.3.1. Bar Graphs  5.3.2. Histograms  5.3.3. Pie Charts  5.3.4. Line Graphs (added)  5.4. Probability  5.4.1. Introduction to Probability  5.4.2. Experimental Probability
  6. Coordinate Geometry  6.1. Cartesian Plane  6.2. Plotting Points  6.3. Coordinate System  6.4. Distance Between Points  6.5. Applications of Coordinate Geometry
  7. Introduction to Graphs  7.1. Linear Graphs  7.2. Applications of Graphs  7.3. Distance-Time Graphs  7.4. Speed-Time Graphs
  8. Comparing Quantities  8.1. Ratio and Proportion  8.2. Percentage  8.2.1. Increase and Decrease Percent  8.2.2. Discount in Percentage  8.2.3. Profit and Loss in Percentage  8.2.4. Simple Interest  8.2.5. Compound Interest
  9. Exponents and Powers  9.1. Laws of Exponents  9.2. Negative Exponents  9.3. Standard Form  9.4. Applications of Exponents
  10. Introduction to Squares and Square Roots  10.1. Properties of Square Numbers  10.2. Finding Square Roots  10.2.1. Prime Factorization Method  10.2.2. Division Method  10.3. Estimating Square Roots
  11. Introduction to Cubes and Cube Roots  11.1. Properties of Cube Numbers  11.2. Finding Cube Roots  11.2.1. Prime Factorization Method in Finding Cube Roots

Grade 8 → Algebra → Identities and Simplification ↓


Using Algebraic Identities


Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. In algebra, these symbols (often represented as letters) stand for numbers and are used to express general relationships through equations and formulas. Algebraic identities are special equations in algebra that are true for all values of the variables within them.

These identities are used extensively to simplify algebraic expressions, solve equations, and understand mathematical structures. Using algebraic identities effectively can make solving complex algebra problems much easier and more intuitive.

What are algebraic identities?

Algebraic identities are equations that are true for any value of the variables involved. They reflect fundamental properties of numbers and operations. Understanding these identities is essential to simplify algebraic expressions or solve algebraic equations.

Some common algebraic identities include:

  • Square of the sum: (a + b)^2 = a^2 + 2ab + b^2
  • Square of the difference: (a - b)^2 = a^2 - 2ab + b^2
  • Product of sum and difference: (a + b)(a - b) = a^2 - b^2
  • Cube expansion: (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3

Importance of algebraic identities

Algebraic identities are important because they help simplify complex expressions and solve polynomial equations. Instead of manually expanding each expression and then simplifying, identities provide a faster way to obtain the result. In addition, some algebraic identities can also be used to factor polynomials or verify solutions.

Understanding common algebraic identities

Square of the sum: (a + b)^2

The square of the sum identity is given by:

(a + b)^2 = a^2 + 2ab + b^2

This equation states that when you have two numbers, say a and b, and you want to find the square of their sum, you can use this identity. Let us understand this with an example:

Consider a = 3 and b = 4:

(3 + 4)^2 = 3^2 + 2*3*4 + 4^2
 = 9 + 24 + 16
 = 49

Instead of calculating (3 + 4)^2 directly, we used identities to break down the steps and simplify each component.

Visual example:

a^2 2AB 2AB b^2

In the above SVG, we can see that the large square is made up of four smaller segments representing a^2, 2ab twice, and b^2, which confirms that (a + b)^2 = a^2 + 2ab + b^2.

Square of the difference: (a - b)^2

The square of the difference identity is:

(a - b)^2 = a^2 - 2ab + b^2

This is similar to the square of the sum, but here b is subtracted. Let's see how this works with numbers:

Consider a = 5 and b = 2:

(5 - 2)^2 = 5^2 - 2*5*2 + 2^2
 = 25 - 20 + 4
 = 9

By using identities, we avoid direct calculations and simplify the terms.

Visual example:

a^2 -2AB -2AB b^2

This SVG shows how the square of the difference appears in the same sections as the square of the sum, but with a negative product, which affects the total area displayed in the identity.

Product of sum and difference: (a + b)(a - b)

This identity is a little different because it involves the product of the sum and difference of the same terms:

(a + b)(a - b) = a^2 - b^2

This is useful for eliminating the middle term and finding the difference of squares:

Consider a = 8 and b = 3:

(8 + 3)(8 - 3) = 8^2 - 3^2
 = 64 - 9
 = 55

This identity quickly simplifies the solution by reducing the number of multiplications.

Cube expansion: (a + b)^3

A common identity involving cubes is the expansion of a binomial to the third power:

(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3

Expanding the three variables into their cubic terms:

Consider a = 2 and b = 1:

(2 + 1)^3 = 2^3 + 3*2^2*1 + 3*2*1^2 + 1^3
 = 8 + 12 + 6 + 1
 = 27

This identity allows us to express cubes in terms of smaller multiples of a and b.

Using algebraic identities to simplify expressions

Now that we know some identities, let's see how they can help us simplify expressions.

Example 1

(x + 5)^2 - (x - 5)^2 Simplify:

(x + 5)^2 = x^2 + 2*5*x + 5^2
 = x^2 + 10x + 25
(x - 5)^2 = x^2 - 2*5*x + 5^2
 = x^2 - 10x + 25
(x + 5)^2 - (x - 5)^2 = (x^2 + 10x + 25) - (x^2 - 10x + 25)
 = x^2 + 10x + 25 - x^2 + 10x - 25
 = 20x

Rather than calculating each expression separately, identification effectively simplifies the process.

Example 2

Simplify (3x + 2)^2 - 4x(3x + 2):

(3x + 2)^2 = 3x^2 + 2*3x*2 + 2^2
 = 9x^2 + 12x + 4
4x(3x + 2) = 12x^2 + 8x
(3x + 2)^2 - 4x(3x + 2) = 9x^2 + 12x + 4 - 12x^2 - 8x
 = -3x^2 + 4x + 4

By using square and distributive identities, the expression is simplified without complex expansions.

Conclusion

Using algebraic identities is a powerful method in algebra that prepares you to solve complex expressions and equations with ease. Recognizing which identities to apply in different algebraic scenarios will greatly enhance problem-solving skills. As you keep practicing, the application of these identities will become second nature, unlocking more complex areas of algebra and higher mathematics.


Grade 8 → 2.2.2


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