←Algebra / Quadratic Identities

Quadratic Identities in Algebra (Algebraic Identities)

🎯 What will you learn in this chapter?

👉 Formulas related to (a + b)² and (a - b)², with explanation.

👉 How to solve problems using the formulas of (a + b)² and (a - b)².

👉 Other useful identities derived from the formulas of (a + b)² and (a - b)².

👉 The origin and proof of important quadratic identities in algebra.

🔹 A quadratic identity is an algebraic identity whose highest power is 2.

🔹 An identity is an equality that remains true for every value of the variables. Here, the variables are a and b in the algebraic expressions.

🔹 These identities are used very often in algebra and in many other branches of mathematics.

🔹 They are mainly used for simplifying algebraic expressions and for factorisation.

🔸 +
(a + b)2= (a + b)×(a + b)
= (a + b)×a + (a + b)×b
= a×a + b×a + a×b + b×b
= a2 + ab + ab + b2
(a + b)2= a2 + 2ab + b2

1. Express as a perfect square.

2. Express as a perfect square.

3. If is a perfect square, fill in the blank.

🔸 +
∵
or,
[ Subtracting 2ab from both sides ]
or,
or,
∴

1. If , find the value of .

2. If , find the value of .

🔸 +
(a - b)2= (a - b)×(a - b)
= (a - b)×a - (a - b)×b
= (a×a - b×a) - (a×b - b×b)
= a×a - b×a - a×b + b×b
= a2 - ab - ab + b2
(a - b)2= a2 - 2ab + b2

1. Express as a perfect square.

2. Simplify using the formula.

3. If is a perfect square, find the value of t.

🔸 +
∵
or,
[ Adding 2ab to both sides ]
or,
or,
∴

1. If , find the value of .

2. If , find the value of .

3. If , find the value of .

🔸 +
∵
or,
or,
∴
🔸 +
∵
or,
or,
∴
🔸 +
∵ ---(i)
∵ ---(ii)
Adding equations (i) and (ii), we get:
Geometric representation of (a+b)² and (a−b)²
or,
∴

1. Simplify

2. If a + b = 5 and a - b = 3, find the value of a2 + b2.

🔸 +
∵ (a + b)(a - b) = (a + b) × (a - b)
∴ (a + b) × (a - b)
= (a + b)×a - (a + b)×b
= (a×a + b×a) - (a×b + b×b)
= a×a + a×b - a×b - b×b
=
=
∴
or,

1. Find the product using the formula: 1216 × 1184.

2. If (x - 3)(x - p) = x2 - 9, find the value of p.

3. Simplify (a + 2b - 3c)2 - (a - 2b + 3c)2.

🔸 +
∵ ---(i)
∵ ---(ii)
Subtracting equation (ii) from equation (i), we get:
Geometric representation of (a+b)² and (a−b)²
or,
∴

1. If a - b = 7 and ab = 8, find the value of a + b.

2. If a + b = 5 and ab = 4, find the value of a - b.

3. If a + b = 7 and a - b = 3, find the value of 8ab(a2 + b2).

🔸 +
∵
Dividing both sides of this equation by 4, we get:
or,

1. Express 265 as the difference of two squares.

2. Express x2 + 5x + 6 as the difference of two squares.

🔸 +
∵
Let x = a + b and y = c. Substituting these values in the above identity, we get:
∴

1. If a + b + c = 15 and a2 + b2 + c2 = 77, find the value of ab + bc + ca.

🔸 +
∵
[ Subtracting from both sides ]
or,
∴

1. If a + b + c = 7 and ab + bc + ca = 12, find the value of (a - b)2 + (b - c)2 + (c - a)2.

2. If x + y + z = 8 and xy + yz + xz = 18, find the value of x2 + y2 + z2.

✅ Quick Revision

🔹 (a + b)² = a² + 2ab + b²

🔹 (a - b)² = a² - 2ab + b²

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

🔹 a² - b² = (a + b)(a - b)

🔹 (a + b)² + (a - b)² = 2(a² + b²)

🔹 4ab = (a + b)² - (a - b)²

🔹 (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

⚠️ Common Mistakes

❌ Do not write (a + b)² = a² + b². The middle term 2ab is necessary.

❌ Do not write (a - b)² = a² - b². The correct identity is a² - 2ab + b².

✅ For difference of squares, use a² - b² = (a + b)(a - b).

✅ In three-variable identities, remember that the pairwise products are ab, bc, and ca.

❓ Frequently Asked Questions

What is a quadratic identity?

A quadratic identity is an algebraic equality involving terms of degree 2 that remains true for all allowed values of the variables.

Which identity is used for the square of a sum?

The identity is (a + b)² = a² + 2ab + b².

Which identity is used for the square of a difference?

The identity is (a - b)² = a² - 2ab + b².

What is the difference of squares formula?

The formula is a² - b² = (a + b)(a - b).