This MCQ module is based on: Square Identities: (a+b)^2, (a-b)^2 and (a+b)(a-b)
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Square Identities: (a+b)^2, (a-b)^2 and (a+b)(a-b)
🎓 Class 9
Mathematics
CBSE
Theory
Ch 4 — Exploring Algebraic Identities
⏱ ~40 min
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This mathematics assessment will be based on: Square Identities: (a+b)^2, (a-b)^2 and (a+b)(a-b)
Targeting Class 9 level in Algebra, with Intermediate difficulty.
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4.1 Introduction — From Patterns to Identities
In earlier chapters, you encountered linear and quadratic polynomials?. You learned how letters can stand in for numbers and capture relationships in compact form. In this chapter, we will explore shortcuts that allow us to multiply, square, and factor algebraic expressions almost instantly — without writing out every step. These shortcuts are called algebraic identities?.
An algebraic identity is an equality that holds true for every value of the variables involved. For example: \[ x + x = 2x \quad\text{(true for any value of } x\text{)} \] This is an identity. By contrast, \(x + 5 = 11\) is an equation — true only for the special value \(x = 6\).
Identity vs Equation
An identity is a statement of equality that is true for all permissible values of the variables (e.g. \((a+b)^2 = a^2 + 2ab + b^2\)). An equation is a statement of equality true only for specific values (e.g. \(2x + 3 = 11\) is true only for \(x = 4\)).A Surprising Pattern in Consecutive Squares
Consider three consecutive square numbers: 25, 36, 49. Subtract twice the middle one from the sum of the smallest and largest: \[ 25 + 49 - (2 \times 36) = 74 - 72 = 2. \] Try with 36, 49, 64: \[ 36 + 64 - (2 \times 49) = 100 - 98 = 2. \] Try with 100, 121, 144: \[ 100 + 144 - (2 \times 121) = 244 - 242 = 2. \] The answer is always 2! Why? This is exactly what algebraic identities can explain. Let the three consecutive integers be \(n-1, n, n+1\). Their squares are \((n-1)^2, n^2, (n+1)^2\). The pattern is: \[ (n-1)^2 + (n+1)^2 - 2n^2 = 2 \] We will prove this in just a few lines once we have our identities.
4.2 The Identity \((a+b)^2 = a^2 + 2ab + b^2\)
Let us derive this identity in two ways — algebraically and geometrically.
Algebraic Proof
By the distributive property: \[ (a+b)^2 = (a+b)(a+b) = a(a+b) + b(a+b) = a^2 + ab + ba + b^2 = a^2 + 2ab + b^2. \] This is true for ANY values of \(a\) and \(b\), so it is an identity.
Identity I: \((a+b)^2 = a^2 + 2ab + b^2\)
Geometric Proof — The Area Model
Consider a square of side \((a+b)\). Its area is \((a+b)^2\). Now divide the square into four parts as shown:
Fig 4.1 — Visual proof: \((a+b)^2 = a^2 + 2ab + b^2\). The big square's area equals the sum of the four pieces.
Adding the four pieces: \(a^2 + ab + ab + b^2 = a^2 + 2ab + b^2\). This must equal the total area \((a+b)^2\). ✓
Worked Examples — Squaring a Sum
Example 1. Find \((103)^2\) without long multiplication.
Write \(103 = 100 + 3\). Apply Identity I: \((100+3)^2 = 100^2 + 2(100)(3) + 3^2 = 10000 + 600 + 9 = 10609\).
Example 2. Expand \((3x + 2y)^2\).
Take \(a = 3x, b = 2y\). \((3x+2y)^2 = (3x)^2 + 2(3x)(2y) + (2y)^2 = 9x^2 + 12xy + 4y^2\).
Example 3. Expand \(\left(2x + \dfrac{1}{x}\right)^2\), assuming \(x \neq 0\).
\(\left(2x + \frac{1}{x}\right)^2 = (2x)^2 + 2(2x)\left(\frac{1}{x}\right) + \left(\frac{1}{x}\right)^2 = 4x^2 + 4 + \frac{1}{x^2}\).
4.3 The Identity \((a-b)^2 = a^2 - 2ab + b^2\)
Following the same logic: \[ (a-b)^2 = (a-b)(a-b) = a(a-b) - b(a-b) = a^2 - ab - ab + b^2 = a^2 - 2ab + b^2. \]
Identity II: \((a-b)^2 = a^2 - 2ab + b^2\)
Geometric Proof — Subtracting from a Square
Start with a square of side \(a\) (area \(a^2\)). Cut off a corner square of side \(b\). The remaining shape can be re-arranged: it equals a square of side \((a-b)\) plus two rectangles of sides \(b \times (a-b)\) plus a square of side \(b\). After careful subtraction: \[ \underbrace{(a-b)^2}_{\text{small square}} = a^2 - 2 \cdot b(a-b) - b^2 = a^2 - 2ab + 2b^2 - b^2 = a^2 - 2ab + b^2.\]
Fig 4.2 — Area decomposition: \(a^2 = (a-b)^2 + 2b(a-b) + b^2\), so \((a-b)^2 = a^2 - 2ab + b^2\).
Worked Examples — Squaring a Difference
Example 4. Find \((98)^2\).
\(98 = 100 - 2\). \((100-2)^2 = 100^2 - 2(100)(2) + 2^2 = 10000 - 400 + 4 = 9604\).
Example 5. Expand \((5p - 3q)^2\).
\(a = 5p, b = 3q\). \((5p-3q)^2 = 25p^2 - 30pq + 9q^2\).
Example 6. If \(a + \frac{1}{a} = 5\), find \(a^2 + \frac{1}{a^2}\).
Square both sides: \(\left(a + \frac{1}{a}\right)^2 = 25\), giving \(a^2 + 2 + \frac{1}{a^2} = 25\), so \(a^2 + \frac{1}{a^2} = 23\).
4.4 The Identity \((a+b)(a-b) = a^2 - b^2\)
Multiply directly: \[ (a+b)(a-b) = a^2 - ab + ab - b^2 = a^2 - b^2. \] This is the famous difference of squares identity.
Identity III: \((a+b)(a-b) = a^2 - b^2\)
Geometric Visualisation
Cut a small \(b \times b\) square out of an \(a \times a\) square. The remaining L-shape has area \(a^2 - b^2\). Slice it along the line shown and re-arrange the two pieces — the result is a rectangle of dimensions \((a+b) \times (a-b)\). So \(a^2 - b^2 = (a+b)(a-b)\). ✓
Fig 4.3 — The L-shape (area \(a^2 - b^2\)) rearranges into a rectangle of dimensions \((a+b) \times (a-b)\).
Worked Examples — Difference of Squares
Example 7. Compute \(105 \times 95\) without a calculator.
Write as \((100+5)(100-5) = 100^2 - 5^2 = 10000 - 25 = 9975\).
Example 8. Factorise \(49x^2 - 64y^2\).
\(49x^2 = (7x)^2,\ 64y^2 = (8y)^2\). Hence \(49x^2 - 64y^2 = (7x+8y)(7x-8y)\).
Example 9. Factorise \(x^4 - 16\).
\(x^4 - 16 = (x^2)^2 - 4^2 = (x^2 + 4)(x^2 - 4) = (x^2 + 4)(x+2)(x-2)\).
Example 10. Show that \((n-1)^2 + (n+1)^2 - 2n^2 = 2\) for any integer n.
Apply Identities I and II: \((n+1)^2 = n^2 + 2n + 1\); \((n-1)^2 = n^2 - 2n + 1\). Sum: \(2n^2 + 2\). Subtract \(2n^2\): \(2\). This proves the consecutive-square pattern observed in 4.1.
Activity: The Paper Square Investigation
L3 ApplyMaterials: Two sheets of coloured craft paper, ruler, scissors, glue.
Predict: Can you visually verify that \((a+b)^2\) is greater than \(a^2 + b^2\)?
- From sheet 1, cut a square of side \(a = 6\) cm and a square of side \(b = 4\) cm. Total area cut: \(36 + 16 = 52\) cm².
- From sheet 2, cut a single square of side \(a + b = 10\) cm. Area: 100 cm².
- Place the two smaller squares onto the larger one, aligned at one corner. They cover only 52 cm². The uncovered area = \(100 - 52 = 48\) cm².
- Predict: this uncovered area should be \(2ab = 2 \cdot 6 \cdot 4 = 48\) cm². ✓
- Cut two rectangles from sheet 1 of size \(a \times b = 6 \times 4\). Their combined area is \(2 \cdot 24 = 48\) cm². They tile the uncovered region exactly.
Insight: The "missing" 48 cm² is exactly the \(2ab\) cross-term in \((a+b)^2 = a^2 + 2ab + b^2\). That cross-term is what makes \((a+b)^2\) bigger than \(a^2 + b^2\). Algebraic identities are not abstract symbol pushing — they are precise statements about area.
Historical Note
The identity \((a+b)^2 = a^2 + 2ab + b^2\) appeared 2,300 years ago in Book II of Euclid's Elements (Proposition 4) — entirely as a geometric statement. The Indian mathematician Aryabhata I (476 CE) used this identity systematically to compute square roots. Around 750 CE, Sanskrit mathematician Śrīdharācārya used the same identity to discover the famous "completing the square" trick, the basis of the quadratic formula. The same identity, in three different cultures, in three different millennia.Competency-Based Questions
Scenario: Mira is tiling her grandmother's square verandah of side \((x+5)\) metres. She wants to leave a smaller central square of side \(x\) metres unpaved (for a flower bed) and tile only the surrounding L-shaped border.
Q1. Express the area to be tiled (the L-shaped border) using an identity, in expanded form.
L3 ApplyAnswer (a) \(10x + 25\). Tiled area = \((x+5)^2 - x^2 = (x^2 + 10x + 25) - x^2 = 10x + 25\) m².
Q2. If \(x = 7\) m, analyse: how many square tiles of side 0.5 m will Mira need (if no breakage)?
L4 AnalyseTiled area = \(10(7) + 25 = 95\) m². Each tile area = \(0.5 \times 0.5 = 0.25\) m². Tiles needed = \(95 / 0.25 = 380\) tiles.
Q3. Mira's brother claims, "If you double x, you double the tiled area." Evaluate this claim using the formula.
L5 EvaluateAt \(x\): area = \(10x + 25\). At \(2x\): area = \(20x + 25\). The ratio is \(\frac{20x+25}{10x+25}\) — not 2 in general. For example, at \(x=5\): areas are 75 vs 125, ratio 1.67 (not 2). The brother's claim is false because of the constant term 25 — the relation is linear but not proportional.
Q4. Design a different verandah whose tiled border area is exactly \(10x + 25\) m² but where the inner unpaved region is NOT square. Provide one specific design with a justification.
L6 CreateOne design: Outer rectangle \((x+5) \times (x+5)\) (area still \((x+5)^2\)) with inner rectangle of dimensions \(x \times x\) — though this is what we started with. Alternative: outer \((x+5)^2\), inner rectangle \(\frac{x}{2} \times 2x\) — same area \(x^2\) inside but different shape. Verification: tiled border = \((x+5)^2 - x^2 = 10x + 25\) m² ✓. Many designs work as long as the inner cutout has area \(x^2\) — ovals, rectangles, even irregular polygons.
Assertion–Reason Questions
Assertion (A): \((a+b)^2 \neq a^2 + b^2\) in general.
Reason (R): The expansion of \((a+b)^2\) contains an additional cross-term \(2ab\).
Reason (R): The expansion of \((a+b)^2\) contains an additional cross-term \(2ab\).
(a) — R correctly justifies why A is true. The cross-term \(2ab\) is non-zero unless \(a = 0\) or \(b = 0\).
Assertion (A): \(99 \times 101 = 9999\).
Reason (R): \((a+b)(a-b) = a^2 - b^2\).
Reason (R): \((a+b)(a-b) = a^2 - b^2\).
(a) — \(99 \times 101 = (100-1)(100+1) = 100^2 - 1 = 9999\). R is the exact identity used.
Assertion (A): \(x^4 - y^4\) factorises as \((x-y)(x+y)(x^2+y^2)\).
Reason (R): Difference of squares: \(p^2 - q^2 = (p+q)(p-q)\).
Reason (R): Difference of squares: \(p^2 - q^2 = (p+q)(p-q)\).
(a) — Apply R twice: \(x^4 - y^4 = (x^2-y^2)(x^2+y^2) = (x-y)(x+y)(x^2+y^2)\).
Frequently Asked Questions
What is the algebraic identity for (a + b)^2 in Class 9?
The identity is (a + b)^2 = a^2 + 2ab + b^2. It says: square the first, plus twice the product, plus square of the second. For example, (x + 5)^2 = x^2 + 10x + 25.
How is (a + b)^2 = a^2 + 2ab + b^2 proved geometrically?
Draw a square of side (a + b). Split each side into segments a and b. The square is partitioned into four pieces: a square of area a^2, a square of area b^2, and two rectangles each of area ab. Adding gives a^2 + 2ab + b^2, which equals the total area (a+b)^2.
How do you use the identity (a - b)^2 = a^2 - 2ab + b^2 to compute 98^2 quickly?
Write 98 = 100 - 2. Then 98^2 = (100 - 2)^2 = 100^2 - 2(100)(2) + 2^2 = 10000 - 400 + 4 = 9604. The identity converts a hard squaring into easy mental arithmetic.
What is the difference of squares identity?
The identity is (a + b)(a - b) = a^2 - b^2. It lets you factorise any expression of the form a^2 - b^2 as (a + b)(a - b). For example, x^2 - 49 = (x + 7)(x - 7).
How do you factorise 4x^2 - 25 using identities?
Recognise 4x^2 = (2x)^2 and 25 = 5^2. Apply a^2 - b^2 = (a + b)(a - b) with a = 2x and b = 5: 4x^2 - 25 = (2x + 5)(2x - 5).
Why are algebraic identities useful in Class 9 Maths?
Identities turn long multiplications into single-step shortcuts, simplify complex algebra, factorise polynomials, and prepare you for quadratic equations, polynomial division and competitive exam techniques in higher classes.
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