Maharashtra Board Class 7 Maths part 2 Chapter 14 Algebraic Formulae Expansion of Squares PDF Download

Algebraic Formulae - Expansion of Squares

Let's Recall

A rectangle ABCD is shown in the figure alongside. Its length is y units and its breadth is 2x units. A square of side x units is cut out from this rectangle. We can use operations on algebraic expressions to find the area of the shaded part. Let us write the area of rectangle ABCD as A(ABCD).

Area of the shaded part = A(ABCD) - A(MNCP)

= 2xy - x²

Area of the shaded part = A(ASPD) + A(SBNM)

= (y - x) × 2x + x²

= 2xy - 2x² + x²

= 2xy - x²

Teacher's Note

When you cut a small square from a big rectangle, you can find the remaining area in two different ways. Both ways give the same answer. This is like dividing land - it does not matter how you divide it, the total land stays the same.

Exam Trick

Always remember: Area by subtraction = Area by addition. If you get different answers using two methods, you made an error.

Points to Remember

The area of a shape can be found in multiple ways.
Subtraction method: Find big area and subtract small area.
Addition method: Add all the small parts together.

The Expanded Form of the Square of a Binomial

The product of algebraic expressions is called their 'expansion' or their 'expanded form'. There are some formulae which help in writing certain expansions. Let's consider some of them.

In the figure alongside, the side of the square PQRS is (x + y).

Therefore, A(PQRS) = (x + y)²

The square PQRS is divided into 4 rectangles: I, II, III, IV

A(PQRS) = Sum of areas of rectangles I, II, III, IV.

Therefore, A(PQRS) = A(Rectangle I) + A(Rectangle II) + A(Rectangle III) + A(Rectangle IV)

(x + y)² = x² + xy + xy + y² = x² + 2xy + y²

Therefore, (x + y)² = x² + 2xy + y²

Now, let us multiply (x + y)² as algebraic expressions.

(x + y)(x + y) = x(x + y) + y(x + y)

= x² + xy + yx + y²

Therefore, (x + y)² = x² + 2xy + y²

The expression obtained by squaring the binomial (x + y) is equal to the expression obtained by finding the area of the square. Therefore, (x + y)² = x² + 2xy + y² is the formula for the expansion of the square of a binomial.

Teacher's Note

You can see the formula (x + y)² = x² + 2xy + y² by drawing a square and dividing it into parts. In India, many builders use this idea when they measure land - a big square plot can be divided into smaller squares and rectangles.

Exam Trick

Remember: (x + y)² always gives three terms - x², y², and 2xy in the middle. Do not forget the 2 in front of xy.

Points to Remember

(x + y)² = x² + 2xy + y²
The formula has three parts: first square, last square, and double the product in the middle.
This works for any values of x and y.

Activity II

In the figure alongside, the square with side a is divided into 4 rectangles, namely, square with side (a - b), square with side b, and two rectangles of sides (a - b) and b.

A(square I) + A(rectangle II) + A(rectangle III) + A(square IV) = A(PQRS)

(a - b)² + (a - b)b + (a - b)b + b² = a²

(a - b)² + 2ab - 2b² + b² = a²

(a - b)² + 2ab - b² = a²

Therefore, (a - b)² = a² - 2ab + b²

Let us multiply the algebraic expressions and obtain the formula.

(a - b)² = (a - b) × (a - b) = a(a - b) - b(a - b)

= a² - ab - ab + b²

= a² - 2ab + b²

Now I Know!

(a + b)² = a² + 2ab + b²

(a - b)² = a² - 2ab + b²

We can verify the formulae by substituting a and b with any numbers.

Thus if a = 5, b = 3, then

(a + b)² = (5 + 3)² = 8² = 64

a² + 2ab + b² = 5² + 2 × 5 × 3 + 3² = 25 + 30 + 9 = 64

(a - b)² = (5 - 3)² = 2² = 4

a² - 2ab + b² = 5² - 2 × 5 × 3 + 3² = 25 - 30 + 9 = 4

Use the given values to verify the formulae for squares of binomials.

(i) a = -7, b = 8

(ii) a = 11, b = 3

(iii) a = 2.5, b = 1.2

Expand.

Example 1: (2x + y)²

= (2x)² + 2(2x) × y + y²

= 4x² + 4xy + y²

Example 2: (x - 2y)²

= x² - 2(x) × 2y + (2y)²

= x² - 4xy + 4y²

Example 3: (51)²

= (50 + 1)²

= 50² + 2 × 50 × 1 + 1²

= 2500 + 100 + 1

= 2601

Example 4: (98)²

= (100 - 2)²

= 100² - 2 × 100 × 2 + 2²

= 10000 - 400 + 4

= 9604

Teacher's Note

Using the formula to find squares of numbers like 98 or 51 is very easy. Instead of multiplying the big number by itself, you break it into 100 - 2 or 50 + 1. This saves time and is faster than normal multiplication.

Exam Trick

For numbers close to 100, use (100 ± small number)². For numbers close to 50, use (50 ± small number)². This makes calculation very fast and easy.

Points to Remember

(a + b)² means (a + b) × (a + b).
(a - b)² means (a - b) × (a - b).
You can find big number squares using this formula instead of long multiplication.
Always check: first term square, last term square, and 2 times the product of both terms.

This is a preview of the first 3 pages. To get the complete book, click below.