Keywords: Vedic Math Algebra Tricks, Algebra Shortcuts, Fast Algebra Methods, Mental Math for Algebra, Solve Algebra Quickly, Algebra Calculation Tricks, Easy Algebra Solutions

Vedic Math Tricks to Solve Algebra

The sixteen Sutras and thirteen Sub-Sutras form the foundation of Vedic algebraic mathematics. This article discusses algebra teachings and shows how to use them to solve algebra problems. Unified mathematics is defined as a state where processes can refer to and understand each other. Vedic formulas are used for addition, subtraction, division, and other mathematical operations.

Algebra seems hard to many students, especially when calculations become lengthy. Vedic Mathematics is an ancient method that makes calculations fast, easy, and interesting. If you want to solve algebra quickly, Vedic Math shortcuts help improve both your speed and accuracy.

Why use Vedic math tricks to solve algebra?

Vedic mathematics not only saves time but also helps us make quick decisions and understand complex problems. Vedic mathematics helps reduce the burden of poor academic performance. Simple mathematics is a common description of Vedic mathematics. But it also gives good techniques for solving mathematical problems. Beyond its utility in arithmetic and algebra, Vedic mathematics is also appreciated for its intuitive and pattern-based approach. The system encourages mental calculation and logical reasoning, promoting a deeper understanding of mathematical concepts instead of relying solely on rote memorisation or repetitive methods.

For instance, the Sutra “Vertically and Crosswise” enables the multiplication of large numbers with remarkable speed and accuracy. Similarly, techniques such as “All from 9 and the Last from 10” simplify complex subtraction problems within seconds.

Vedic methods in education

Students and educators alike have found Vedic methods particularly helpful in exam preparation and competitive test scenarios. Vedic math is also used as a brain development tool for young students, helping to build confidence and enhance concentration. The clarity and elegance of the system appeal to people from various educational backgrounds, making math more accessible and less intimidating. Additionally, it bridges gaps between traditional learning and modern education, preserving ancient wisdom while addressing contemporary academic challenges. As a result, Vedic mathematics is gaining global recognition for its effectiveness and versatility.

What Is Vedic Mathematics?

Vedic Maths is an ancient system of mathematics. It simplifies complex mathematical operations using unique techniques that make the problem more manageable and easier to calculate mentally with formulas known as the sutras. These methods provide fast, efficient, and innovative ways to solve problems that traditionally require lengthy calculations.

With a surge in global interest in improving mental math skills, Vedic Maths is now a preferred choice for students, educators, and professionals.

Its versatility and ability to break down intimidating problems make it much easier to do it just mentally with a little practice and experience.

Vedic Mathematics is a collection of Techniques/Sutras to solve mathematical arithmetic in an easy and faster way. It consists of 16 Sutras (Formulae) and 13 sub-sutras (Sub Formulae), which can be used for problems involved in arithmetic, algebra, geometry, calculus, and conics.

How Vedic Math Helps in Algebra

Vedic Math makes algebra easier by simplifying long and complicated steps. It provides smart shortcuts that help students solve expressions and equations quickly without getting confused.

Here’s how it helps:

Faster calculations: Reduces time spent on lengthy operations.

Less confusion: Clear methods make problems easier to understand.

Better accuracy: Fewer steps mean fewer mistakes.

Mental math improvement: Strengthens quick thinking and problem-solving skills.

Simplifies complex expressions: Helps break down algebraic terms easily.

Boosts confidence: Students feel more comfortable solving algebra problems.

In short, Vedic Math turns algebra into a simpler, quicker, and more enjoyable subject.

Vedic Math Tricks to Solve Algebra

1. Vertically & Crosswise Method (Fast Multiplication)

In algebra, we often need to multiply algebraic expressions like 

(axe + b)(cx + d). In the traditional method, this takes 3–4 steps. But in Vedic Math, the vertically and crosswise method answers in just seconds.

We use the vertical and crosswise methods for two-term (axe + b)(cx + d) algebraic expression multiplication.

General Algebraic Form (2 Terms × 2 Terms): (ax + b)(cx + d)

Process: (Left → Right)

1.      Vertical multiplication of 1st digits(left) of 2 numbers.

2.      Crosswise multiplication and  addition

3.      Vertical multiplication of last digits(right) of 2 numbers.

Formula: (ax + b)(cx + d) = acx² + (ad + bc)x + bd

Example 1

Solve: (x + 5)(x + 7)

Solution: We solve step by step

Step 1: Multiplying vertically on the left

 x × x = x²

Step 2: We multiply crosswise and add the two results

 = x × 7 + 5 × x

= 7x+5x

=12x

Step 3: Multiplying vertically on the right

5 × 7 = 35

Combine steps 1, 2, and 3

=x² + 12x + 35   Ans

Example 2

Solve: (2x + 5)(3x + 1)

Solution: We solve step by step

Step 1: Multiplying vertically on the left

2x × 3x = 6x²,

Step 2: We multiply crosswise and add the two results

 =2x × 1 + 5 × 3x

= 2x+15x

=17x

Step 3: Multiplying vertically on the right

5 × 1 = 5.

Combine steps 1, 2, and 3

=6x² + 17x + 5   Ans

Example 3

Solve: (10x - 5)(3x -2):

Solution: We solve step by step

Step 1: Multiplying vertically on the left

10x × 3x = 30x²,

Step 2: We multiply crosswise and add the two results

 =10x × (-2)  + (-5)  × 3x

= -20x-15x

=-35x

Step 3: Multiplying vertically on the right

(-5) × (-2) = 10

Combine steps 1,  2, and 3

=30x² - 35x + 10   Ans

Example 4

Solve: (4x + 5)(2x +3)

Solution: We solve by using a direct formula

Formula: (ax + b)(cx + d) = acx² + (ad + bc)x + bd

Here a=4, b=5, c=2, d=3

(4x + 5)(2x +3)=(4)(2)x²+ ((4)(3)+(5)(2))x+(5)(3)

=8x² +(12+10)x+15

=8x² +22x+15  Ans

Example 5

Solve: (20x + 1)(3x +2)

Solution: We solve by using a direct formula

Formula: (ax + b)(cx + d) = acx² + (ad + bc)x + bd

Here a=20, b=1, c=3, d=2

(20x + 1)(3x +2)=(20)(3)x²+ ((20)(2)+(1)(3))x+(1)(2)

=60x² +(40+3)x+2

=60x² +43x+2

General Algebraic Form (3 Terms × 3 Terms) (ax² + bx + c)(dx² + ex + f)

Process: (Left → Right)

1.      Vertical multiplication of 1st digits of 2 numbers.

2.      Crosswise addition of 1st & 2nd digits of numbers.

3.      Crosswise addition of all 3 digits of both numbers.

4.      Crosswise addition of the last 2 digits of 2 numbers.

5.      Vertical multiplication of the last digits of 2 numbers.

Formula:

(ax² + bx + c)(dx² + ex + f) = adx⁴ + (ae + bd)x³ + (af + be + cd)x² + (bf + ce)x + cf

Example 1:

Solve: (x² + 3x + 5)(2x² + 4x + 6)

Solution: We solve step by step

Start from (Left → Right):

Step 1: Vertical multiplication of 1st digits of 2 numbers.

x² × 2x² = 2x⁴

Step 2: Crosswise addition of 1st & 2nd digits of numbers.

   =x² × 4x + 3x × 2x²

   = 4x³ + 6x³

   = 10x³

Step 3: Crosswise addition of all 3 digits of both numbers.

    =x² × 6 + 3x × 4x + 5 × 2x²

   = 6x² + 12x² + 10x²

   = 28x²

Step 4: Crosswise addition of the last 2 digits of 2 numbers.

    = 3x × 6 + 5 × 4x

   = 18x + 20x

   = 38x

Step 5: Vertical multiplication of the last digits of 2 numbers.

 5 × 6 = 30

Combine all steps

=2x⁴ + 10x³ + 28x² + 38x + 30   Ans

Example 2:

Solve: (x² + 2x + 4)(3x² + 5x + 1)

Solution: We solve step by step

Start from (Left → Right):

Step 1: Vertical multiplication of 1st digits of 2 numbers.

=x² × 3x² = 3x⁴

Step 2: Crosswise addition of 1st & 2nd digits of numbers.

=x² × 5x + 2x × 3x²

   = 5x³ + 6x³

   = 11x³

Step 3: Crosswise addition of all 3 digits of both numbers.

=x² × 1 + 2x × 5x + 4 × 3x²

   = x² + 10x² + 12x²

   = 23x²

Step 4: Crosswise addition of the last 2 digits of 2 numbers.

=2x × 1 + 4 × 5x

   = 2x + 20x

   = 22x

Step 5: Vertical multiplication of the last digits of 2 numbers.

4 × 1 = 4

Combine all steps

=3x⁴ + 11x³ + 23x² + 22x + 4  Ans

Example  3:

Solve: (x² + 2x + 3)(2x² + 4x + 5)

Solution: We solve directly by using the formula

Formula:

(ax² + bx + c)(dx² + ex + f) = adx⁴ + (ae + bd)x³ + (af + be + cd)x² + (bf + ce)x + cf

Here a =1, b=2, c=3, d=2, e=4, f=5

= (1×2) x⁴ + (1×4 + 2×2) x³ + (1×5 + 2×4 + 3×2)x² + (2×5 + 3×4)x + (3×5)

=2 x⁴ + (4 + 4) x³ + (5 + 8 + 6)x² + (10 + 12)x + 15

= 2x⁴ + 8x³ + 19x² + 22x + 15  Ans

Example 4:

Solve: (3x² + 5x + 2)(x² + 6x + 7)

Solution: We solve directly by using the formula

Formula:

(ax² + bx + c)(dx² + ex + f) = adx⁴ + (ae + bd)x³ + (af + be + cd)x² + (bf + ce)x + cf

Here a =3, b=5, c=2, d=1, e=6, f=7

= (3×1) x⁴ + (3×6 + 5×1)x³ + (3×7 + 5×6 + 2×1)x² + (5×7 + 2×6)x + (2×7)

=3x⁴ + (18 + 5)x³ + (21 + 30 + 2)x² + (35+ 12)x + 14

= 3x⁴ + 23x³ + 53x² + 47x + 14  Ans

2. Nikhilam Method (Fast Squaring Trick)

The squares of (x + a)² or (x − a)² are solved very quickly in Vedic Math.

Vedic trick: These form (x + a)² or (x − a)² square of 1st and last digit and double of 2nd digit/number

Example 1:

Find (x + 12)²

Solution: We solve by using the Vedic Trick:

Square of 1st and last number

(x)²=x²  and 12² = 144

Double of 2nd number 12 × 2 = 24

= x² + 24x + 144  Ans

Example 2:

Find (x + 5)²

Solution: We solve by using the Vedic Trick:

Square of 1st and last number

(x)²=x²  and (5)² = 25

Double of 2nd number 5 × 2 = 10

= x² + 10x + 25  Ans

Example 3:

Find (x -6)²

Solution: We solve by using the Vedic Trick:

Square of 1st and last number

(x)²=x²  and (-6)² = 36

Double of 2nd number (-6) × 2 = -12

= x² -12x +36  Ans

3. Easy Factorisation Trick Using Vedic Sutra

We can apply Vedic maths tricks to factorise the quadratic equation

Factorisation using Vedic Mathematics is done by using 2 Sutras. We use a combination of 2 sutras.

1.        Anurupyena(Proportionality).

2.       Adyamadyenantyamantya (1st by 1st and last by last

In       Anurupyena, we split the middle term (coefficient of x) of the quadratic equation into 2 terms such that the Proportion/Ratio of the coefficient of the x² term to 1st coefficient of the x term = the Ratio of 2nd coefficient of the x term to the constant term. That ratio of the 1st 2 coeff is one of the roots of the equation.

In Adyamadyenantyamantya (Commonly called Adyamadyena), we divide the first term’s coefficient of the equation by 1st term of the factor obtained above and the last term of the equation by the last term of the same factor.

Let’s understand this trick with an algorithm, with an example

Example 1:

Factorise: 2x² + 5x -3

Solution: Anurupyena, Split the middle terms coefficient (5) into 2 parts such that the coefficient of x² term to the 1st coeff of x term = Ratio of the 2nd coefficient of x term to the constant term.

Hence split it in 6 and -1 (2/6 = -1/-3) => 2x² + 6x –x -3

So 1st factor: x+3 (2:6)

 Adyamadyenantyamantya: Divide the first coefficient (2) of eq by 1st term of the factor(1), and divide the last term of eq (-3) by 2nd term of the factor (3)

So 2nd factor: 2x-1

And get the factors of 2x² + 5x -3 is (x +3)(2x-1)

Example 2:

Factorize: 3x² + 11x +6

Solution: Anurupyena, Split the middle terms coeff(11) into 2 parts such that

The coefficient of the x² term to the 1st coefficient of the x term = the Ratio of the 2nd coefficient of the x term to the constant term.

Hence split it in 9 and 2 (3/9 = 1/3) => 3x² + 9x +2x +6

So 1st factor: x+3 (3:9)

 Adyamadyenantyamantya: Divide the first term’s coefficient (3) of eq by 1st term of the factor(1), and divide the last term of eq (6) by 2nd term of the factor (3)

So 2nd factor: 3x+2

And get the factors of 3x² + 11x +6 is (x +3)(3x+2)

Example 3:

Factorize: 2x²+ 7x + 3

Solution: Anurupyena: Split the middle terms coeff(7) into 2 parts such that

The coefficient of the x² term to the 1st coefficient of the x term = the Ratio of the 2nd coefficient of the x term to the constant term.

Hence split it in 6 and 1 (2/6 = 1/3) => 2x²+ 6x+x + 3

So 1st factor: x+3 (2:6)

Adyamadyenantyamantya: Divide the first term’s coefficient (2) of eq by 1st term of factor(1) and divide the last term of eq (3) by 2nd term of factor (3)

So 2nd factor: 2x+1

And get the factors of 2x²+ 7x + 3is (x +3)(2x+1)

Traditional Algebra vs Vedic Math Method

Traditional Algebra involves step-by-step calculations that are usually slow. You have to write down many intermediate steps and carefully apply formulas, which can make it confusing and harder for kids to understand nd.

Vedic Math Method uses simple tricks and patterns that make calculations very fast. Because of its straightforward approach, it is easy to learn, fun to practice, and especially helpful for kids to build confidence in math.

Tips to Practice Vedic Algebra

Here are 6 tips for learning Vedic tricks for algebra :

1. Start with the basics

Vedic mathematics is based on 16 Sutras or formulas used to solve mathematical problems. First, you must understand these formulas and how they work before you start solving problems. This will give you a solid foundation to build upon as you progress.

2. Practice regularly

Like with any skill, the more you practice Vedic tricks, the better you will become. Setting aside time each day to practice and continue building your skills is essential. This helps you retain what you have learned and allows you to identify areas where you need more improvement.

Practice Vedic Maths daily for just 20 minutes to see the best results

3. Use visual aids

Visual aids such as diagrams, illustrations, and charts can help understand the concepts of Vedic tricks. Seeing the methods illustrated visually can make them easier to understand and remember.

4. Study the examples

Studying examples of how to solve problems using Vedic tricks can help you understand how the methods work in real-life situations. This can give you a better understanding of the practical application of the methods and can help you see how they can be used to solve various mathematical problems.

5. Work with others

Learn from a teacher: Consider finding a teacher or tutor specialising in Vedic tricks. A teacher can provide guidance, answer questions, and give you feedback on your progress. Having someone to help you navigate the learning process can be very beneficial and help you stay motivated and on track.

6. Be patient

Learning Vedic tricks takes time and patience, so keep going if you don’t see immediate results. The methods and concepts may take some time to understand and internalise, but you will improve with consistent practice and persistence. It’s essential to be patient with yourself and to focus on the progress that you are making, no matter how small it may seem at first.

Conclusion

In my view, Vedic Math is an exceptional tool that transforms the way students approach algebra. It makes complex problems not only faster to solve but also easier to understand and more interesting to work with. By regularly practising these clever shortcuts and techniques, students can significantly increase their calculation speed, reduce errors, and enhance their problem-solving skills. Over time, even the most challenging algebra questions can be solved in just a few simple steps, boosting confidence and making learning mathematics a much more enjoyable experience.

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