Quadratic Identities in Algebra: Solved Problems
🔸 01. Express as a perfect square.
∵
=
=
[ ∵ (a + b)2 = a2 + 2ab + b2 ]
🔸 02. Express 4x2 + 12xy + 9y2 as a perfect square.
∵ 4x2 + 12xy + 9y2
= (2x)2 + 12xy + (3y)2
= (2x)2 + 2.2x.3y + (3y)2
= (2x + 3y)2
[ ∵ (a + b)2 = a2 + 2ab + b2 ]
🔸 03. If is a perfect square, fill in the missing term.
∵ 4x4 + ...... + 49y2
= (2x2)2 + ...... + (7y)2
Using (a + b)2 = a2 + 2ab + b2, the middle term is 2ab.
that is, 2.2x2.7y
= 28x2y
🔸 04. If , find .
∵
or,
[ ∵ Squaring both sides. ]
or,
or,
or,
or,
🔸 05. If , find .
∵
or,
[ ∵ Multiplying both sides by 2. ]
or,
or,
or,
[ ∵ Squaring both sides. ]
or,
or,
or,
or,
🔸 06. Express as a perfect square.
∵
=
=
=
[ ∵ (a + b)2 = a2 + 2ab + b2 ]
🔸 07. Simplify using an identity.
∵ (2q − 3z)2 − 2(2q − 3z)(q - 3z) + (q - 3z)2
Let a = (2q − 3z) and b = (q − 3z).
or, a − b = (2q − 3z) − (q − 3z)
or, a − b = 2q − 3z − q + 3z
or, a − b = q
∴ (2q − 3z)2 − 2(2q − 3z)(q − 3z) + (q − 3z)2
= a2 − 2ab + b2
= (a − b)2
[ ∵ (a − b)2 = a2 − 2ab + b2 ]
= q2
🔸 08. If is a perfect square, find t.
∵ 4x2 + tx + 9
=
=
=
For the expression to be a perfect square, .
or, t = ± 12
∴ t = 12 or t = − 12
🔸 09. If , find .
∵
or,
[ ∵ Squaring both sides. ]
or,
or,
or,
or,
🔸 10. If , find .
∵
or,
[ ∵ Squaring both sides. ]
or,
or,
or,
or,
[ ∵ Squaring both sides again. ]
or,
or,
or,
or,
or,
🔸 11. If , find .
∵
or,
[ ∵ Squaring both sides. ]
or,
or,
or,
or,
🔸 12. Simplify .
∵
=
[ ∵ (a + b)2 = (a − b)2 + 4ab ]
= 4
🔸 13. If a + b = 5 and a − b = 3, find a2 + b2.
∵ (a + b)2 + (a − b)2 = 2(a2 + b2)
∴ 2(a2 + b2) = (a + b)2 + (a − b)2
∴ 2(a2 + b2) = 52 + 32
∴ 2(a2 + b2) = 25 + 9
∴ 2(a2 + b2) = 34
∴ a2 + b2 = 17
🔸 14. Find the product 1216 × 1184 using an identity.
∵ 1216 × 1184
= (1200 + 16) × (1200 − 16)
= (1200 + 16) × (1200 − 16)
= (1200)2 − (16)2
[ ∵ (a + b)(a − b) = a2 − b2 ]
= 1440000 − 256
= 14,39,744
🔸 15. If the identity (x − 3)(x + p) = x2 − 9 is true for all x, find p.
∵ (x − 3)(x + p) = x2 − 9
or, (x − 3)(x + p) = x2 − 32
[ ∵ Using (a + b)(a − b) = a2 − b2. ]
∴ p = 3
🔸 16. Simplify (a + 2b − 3c)2 − (a − 2b + 3c)2.
∵ (a + 2b − 3c)2 − (a − 2b + 3c)2
= {a + (2b − 3c)}2 − {a − (2b − 3c)}2
= {a − (2b − 3c)}2 + 4a(2b − 3c) − {a − (2b − 3c)}2
[ ∵ (a + b)2 = (a − b)2 + 4ab ]
= 4a(2b − 3c)
👇 Practice Yourself
🔹 17. If and , find x2 - y2.
🔹 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.
🔹 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.
