Algebra - Important Concepts & Solved Questions

“Algebra is the branch of mathematics dealing with symbols and the rules for manipulating them.”

Type 1: Algebraic Identities

  • Q1: (a + b)2 = ?
    Ans: a2 + 2ab + b2
    Explanation: It is a standard identity used for squaring binomials.
  • Q2: Simplify: (x + 3)2 - (x - 3)2
    Ans: 12x
    Explanation: Use identity a2 - b2 = (a + b)(a - b). Here a = x+3, b = x-3
  • Q3: Evaluate: (2a + b)2 - 4ab
    Ans: 4a2 + b2
    Explanation: Expand and simplify using the identity (a + b)2 = a2 + 2ab + b2
  • Q4: (a - b)2 = ?
    Ans: a2 - 2ab + b2
    Explanation: Identity for square of binomial difference.
  • Q5: a3 + b3 = ?
    Ans: (a + b)(a2 - ab + b2)
    Explanation: Standard identity for sum of cubes.

Type 2: Factorization

  • Q6: Factorize: x2 - 9
    Ans: (x + 3)(x - 3)
    Explanation: Use identity a2 - b2 = (a + b)(a - b)
  • Q7: Factorize: x2 + 7x + 12
    Ans: (x + 3)(x + 4)
    Explanation: Find two numbers whose product is 12 and sum is 7.
  • Q8: Factorize: x2 - 5x + 6
    Ans: (x - 2)(x - 3)
    Explanation: Factors of 6 are 2 and 3, and their sum is 5.
  • Q9: Factorize: a3 - b3
    Ans: (a - b)(a2 + ab + b2)
    Explanation: Difference of cubes identity.
  • Q10: Factorize: x3 + 3x2 + 3x + 1
    Ans: (x + 1)3
    Explanation: Recognize the identity a3 + 3a2b + 3ab2 + b3

Type 3: Simplification

  • Q11: Simplify: (x + 1)(x - 1)
    Ans: x2 - 1
    Explanation: Use identity a2 - b2
  • Q12: Simplify: (x + 2)2 + (x - 2)2
    Ans: 2x2 + 8
    Explanation: Expand both squares and add them.
  • Q13: Simplify: (a + b)3 + (a - b)3
    Ans: 2a3 + 6ab2
    Explanation: Use identity: a3 + b3 = (a + b)3 - 3ab(a + b)
  • Q14: Simplify: x2 - (x + 1)(x - 1)
    Ans: 1
    Explanation: Apply the identity and subtract.
  • Q15: Simplify: (a + b)2 - (a - b)2
    Ans: 4ab
    Explanation: Use the expansion identities and subtract.

Type 4: Linear Equations

  • Q16: Solve: 2x + 3 = 7
    Ans: x = 2
    Explanation: Subtract 3 from both sides, then divide by 2.
  • Q17: Solve: 5x - 2 = 3x + 4
    Ans: x = 3
    Explanation: Collect like terms and solve for x.
  • Q18: Solve: 4x - 7 = 2x + 1
    Ans: x = 4
    Explanation: Bring variables on one side and constants on the other.
  • Q19: Solve: 6x + 5 = 3x + 14
    Ans: x = 3
    Explanation: Rearrange and solve.
  • Q20: Solve: x/2 + 3 = 5
    Ans: x = 4
    Explanation: Subtract 3 and multiply both sides by 2.

Type 5: Miscellaneous Algebra

  • Q21: If a + b = 10 and ab = 21, find a2 + b2
    Ans: 58
    Explanation: Use identity: a2 + b2 = (a + b)2 - 2ab = 100 - 42 = 58
  • Q22: If x + 1/x = 5, find x2 + 1/x2
    Ans: 23
    Explanation: Use identity: (x + 1/x)2 = x2 + 1/x2 + 2
  • Q23: If x + 1/x = 2, find x3 + 1/x3
    Ans: 2
    Explanation: Use identity: (x + 1/x)3 = x3 + 1/x3 + 3(x + 1/x)
  • Q24: If a + b = 12 and ab = 35, find a3 + b3
    Ans: 1728
    Explanation: Use identity a3 + b3 = (a + b)3 - 3ab(a + b)
  • Q25: If a - b = 2 and ab = 15, find a2 + b2
    Ans: 34
    Explanation: Use identity a2 + b2 = (a - b)2 + 2ab

Additional Practice

  • Q26: Solve for x: x2 = 49
    Ans: x = ±7
  • Q27: If x = 2, find value of x3 + 3x2 + 3x + 1
    Ans: 27
    Explanation: Identity of (x + 1)3
  • Q28: If x + y = 4 and x - y = 2, find x and y
    Ans: x = 3, y = 1
    Explanation: Solve system of equations using elimination method.
  • Q29: Simplify: a2 - (a - b)2
    Ans: 2ab - b2
  • Q30: Solve: 3x/2 = 12
    Ans: x = 8
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