Visual Mathematics Laboratory - Interactive Algebra Learning

Math Mantra Nepal (MMN) – Visual Algebra Lab | Geometrical Proof of Square of a Plus B and A Minus B

Geometrical Proof of Square of A Plus B and A Minus B

Visual Algebra Lab by Kailash Pahari (MMN) | Interactive Proofs

a = 6

b = 2

Show Numerical Values (अङ्क देखाउनुहोस्)

Select Formula / सुत्र छान्नुहोस्

MMN Visual Lab Note:

अङ्कहरू हेर्न माथिको ‘Show Numerical Values’ चेकबक्समा क्लिक गर्नुहोस्। $a$ र $b$ को मान स्लाइडरबाट परिवर्तन गर्न सकिन्छ।

Interactive Geometrical Proof

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Math Mantra Nepal (MMN) Visual Laboratory by Kailash Pahari

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Hover or tap any shape on the canvas to inspect its dimensions and area.

Proof Explanation

1. Fundamental Algebraic Formulas Summary

Square of Sum

$(a + b)^2$

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

The area of a square with side length $(a+b)$ equals the sum of two squares $a^2, b^2$ and two rectangles $ab$.

Square of Difference

$(a – b)^2$

$(a – b)^2 = a^2 – 2ab + b^2$

Derived by subtracting two strips of $b(a-b)$ and a corner square $b^2$ from a large square $a^2$.

Difference of Squares

$a^2 – b^2$

$a^2 – b^2 = (a + b)(a – b)$

Cutting $b^2$ from $a^2$ and rearranging the remaining rectangle forms a rectangle of sides $(a+b)$ and $(a-b)$.

Sum of Squares

$a^2 + b^2$

$a^2 + b^2 = (a+b)^2 – 2ab$

Alternative Form: $a^2 + b^2 = (a-b)^2 + 2ab$. Frequently used in algebraic value evaluations.

2. Curriculum Importance & Practical Applications

Target Grade Levels & Concepts

Understanding the a plus b whole square and a minus b whole square proof forms the core foundation of school algebra in Grade 6, Grade 7, Grade 8 (BLE), Grade 9, and Grade 10 (SEE Examinations) in Nepal and globally. The geometrical proof of square of aplus b and a minus b developed by Kailash Pahari at MMN enables students to visualize algebraic expansions through area models.

Real-World & Advanced Applications

These identities are utilized extensively in polynomial factorization, mental arithmetic shortcuts (e.g., $98^2 = (100 – 2)^2$), calculus integrations, coordinate geometry distance calculations, and physics formulas involving square vector magnitudes.

3. Step-by-Step Worked Examples

Example 1: Difference of Squares Factorization

Factorize the expression: $4x^2 – y^2$

Solution:

Rewriting as perfect squares: $(2x)^2 – (y)^2$

Applying $a^2 – b^2 = (a + b)(a – b)$ where $a = 2x, b = y$:

$= (2x + y)(2x – y)$

Example 2: Fractional Difference of Squares Algebraic Fractions

Factorize: $25x^2 – \frac{1}{49y^2}$

Solution:

Expressing as squares: $(5x)^2 – \left(\frac{1}{7y}\right)^2$

Using $a^2 – b^2 = (a + b)(a – b)$:

$= \left(5x + \frac{1}{7y}\right)\left(5x – \frac{1}{7y}\right)$

Example 3: Binomial Expansions Expansion

Expand $(3x + 2y)^2$ and $(3x – 2y)^2$

(i) Expansion of $(3x + 2y)^2$:

$= (3x)^2 + 2(3x)(2y) + (2y)^2$

$= 9x^2 + 12xy + 4y^2$

(ii) Expansion of $(3x – 2y)^2$:

$= (3x)^2 – 2(3x)(2y) + (2y)^2$

$= 9x^2 – 12xy + 4y^2$

Example 4: Reciprocal Term Expansion Reciprocals

Expand: $\left(2x – \frac{1}{2x}\right)^2$

Solution:

Using $(a – b)^2 = a^2 – 2ab + b^2$:

$= (2x)^2 – 2(2x)\left(\frac{1}{2x}\right) + \left(\frac{1}{2x}\right)^2$

Canceling common factors in middle term $2(2x)\left(\frac{1}{2x}\right) = 2$:

$= 4x^2 – 2 + \frac{1}{4x^2}$

4. Special Cases & Value Evaluation Problems

Dual Forms of $a^2 + b^2$

The sum of squares $a^2 + b^2$ can be expressed in two distinct ways based on given information:

Format 1 (When sum $a+b$ is known):

$a^2 + b^2 = (a + b)^2 – 2ab$

Format 2 (When difference $a-b$ is known):

$a^2 + b^2 = (a – b)^2 + 2ab$

Evaluation Problem Solution

Given $a + b = 5$ and $ab = 6$, find the value of $a^2 + b^2$.

Given: $a + b = 5$, $ab = 6$

Method 1 (Using Formula):

$a^2 + b^2 = (a + b)^2 – 2ab$

$a^2 + b^2 = (5)^2 – 2(6)$

$a^2 + b^2 = 25 – 12$

$\therefore a^2 + b^2 = 13$

Visual Mathematics Laboratory – Interactive Algebra Learning

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