Algebra / Quadratic IdentitiesQuadr ...

Quadratic Identities in Algebra

🎯 What will you learn at the end of this chapter?

👉 The formulas related to (a + b)² and (a − b)², with explanations.

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

👉 Other important formulas derived from (a + b)² and (a − b)².

👉 The derivation and proof of quadratic identities in algebra.

🔹 In algebra, an identity whose highest power is 2 is called a quadratic identity.

🔹 An identity is an equality that remains true for all permissible values of its variables. Here a and b are variables.

🔹 These identities are widely used in algebra and in other branches of mathematics.

🔹 They are commonly used to simplify algebraic expressions, evaluate expressions, and factorise them.

🔸 +
(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
🔸 +
∵
or,
[ ∵ Subtracting 2ab from both sides. ]
or,
or,
∴
🔸 +
(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
🔸 +
∵
or,
[ ∵ Adding 2ab to both sides. ]
or,
or,
∴

✍ Practice Problems Based on the Identities Above

🔹 01. Express as a perfect square.

🔹 02. Express 4x2 + 12xy + 9y2 as a perfect square.

🔹 03. If is a perfect square, fill in the missing term.

🔹 04. If , find .

🔹 05. If , find .

🔹 06. Express as a perfect square.

🔹 07. Simplify using an identity.

🔹 08. If is a perfect square, find t.

🔹 09. If , find .

🔹 10. If , find .

🔹 11. If , find .

👉 Click here to see the step-by-step solutions.

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

✍ Practice Problems Based on the Identities Above

🔹 12. Simplify .

🔹 13. If a + b = 5 and a − b = 3, find a2 + b2.

🔹 14. Find the product 1216 × 1184 using an identity.

🔹 15. If the identity (x − 3)(x + p) = x2 − 9 is true for all x, find p.

🔹 16. Simplify (a + 2b − 3c)2 − (a − 2b + 3c)2.

🔹 17. If and , find x2 - y2.

👉 Click here to see the step-by-step solutions.

🔸 +
∵ ---(i)
∵ ---(ii)
Subtracting equation (ii) from equation (i), we get:
Geometric representation of (a+b)² and (a−b)²
or,
∴
🔸 +
∵
Dividing both sides of this equation by 4, we get:
or,

✍ Practice Problems Based on the Identities Above

🔹 18. If a − b = 7, ab = 8, and a, b > 0, find a + b.

🔹 19. If a + b = 5, ab = 4, and a > b, find a − b.

🔹 20. If a + b = 7 and a − b = 3, find 8ab(a2 + b2).

🔹 21. Express 265 as the difference of two squares.

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

🔸 +
∵
Let x = a + b and y = c. Substituting these values into the identity above, we get:
∴
🔸 +
∵
[ ∵ Subtracting from both sides. ]
or,
∴

✍ Practice Problems Based on the Identities Above

🔹 23. If a + b + c = 15 and a2 + b2 + c2 = 77, find ab + bc + ca.

🔹 24. If a + b + c = 7 and ab + bc + ca = 12, find (a − b)2 + (b − c)2 + (c − a)2.

🔹 25. If x + y + z = 8 and xy + yz + xz = 18, find x2 + y2 + z2.

🧠 Quick Revision

⭐ (a + b)² = a² + 2ab + b²

⭐ (a + b)² = (a − b)² + 4ab

⭐ (a − b)² = a² − 2ab + b²

⭐ (a − b)² = (a + b)² − 4ab

⭐ a² + b² = (a − b)² + 2ab

⭐ a² + b² = (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)

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