Solutions by Factorization MCQs Quiz | Class 10

Welcome to the Class X Mathematics (Code 041) quiz on Unit II: Algebra, specifically ‘Solutions by Factorization’. This quiz covers the method, examples, and checks for solving quadratic equations by factorization. After attempting all 10 questions, click ‘Submit Quiz’ to see your score and then ‘Download Answer PDF’ for a detailed answer sheet.

Understanding Solutions by Factorization

Quadratic equations, typically in the form `ax^2 + bx + c = 0` (where a, b, c are real numbers and a is not equal to 0), are fundamental in mathematics. Finding their ‘solutions’ or ‘roots’ means determining the values of ‘x’ that satisfy the equation. Factorization is a powerful and direct method for solving many quadratic equations.

The Method of Factorization (Step-by-Step)

  1. Standard Form: Ensure the quadratic equation is written in the standard form `ax^2 + bx + c = 0`. If not, rearrange the terms.
  2. Factorize the Expression: The goal is to express the quadratic polynomial `ax^2 + bx + c` as a product of two linear factors, e.g., `(px + q)(rx + s)`.
    • Splitting the Middle Term: For a general quadratic, find two numbers whose product is `ac` and whose sum is `b`. Use these numbers to split the middle term `bx` into two terms, then group the terms and factor by grouping.
    • Difference of Squares: If the equation is in the form `x^2 – k^2 = 0`, it can be factored as `(x – k)(x + k) = 0`.
    • Common Factor: If `c = 0` (e.g., `ax^2 + bx = 0`), factor out ‘x’ to get `x(ax + b) = 0`.
  3. Apply the Zero Product Rule: Once the equation is factored into the form `(factor 1)(factor 2) = 0`, the Zero Product Rule states that for the product to be zero, at least one of the factors must be zero. So, set each factor equal to zero: `factor 1 = 0` or `factor 2 = 0`.
  4. Solve for x: Solve the resulting linear equations to find the values of x. These are the solutions or roots of the quadratic equation.

Example: Solving `x^2 – 7x + 12 = 0`

Let’s walk through an example:

  1. Standard Form: The equation `x^2 – 7x + 12 = 0` is already in standard form. Here, `a=1`, `b=-7`, `c=12`.
  2. Factorize: We need two numbers whose product is `ac = 1 * 12 = 12` and whose sum is `b = -7`. The numbers are -3 and -4.
    • Split the middle term: `x^2 – 3x – 4x + 12 = 0`
    • Group terms: `(x^2 – 3x) – (4x – 12) = 0`
    • Factor out common monomials: `x(x – 3) – 4(x – 3) = 0`
    • Factor out the common binomial: `(x – 3)(x – 4) = 0`
  3. Zero Product Rule: Set each factor to zero:
    • `x – 3 = 0`
    • `x – 4 = 0`
  4. Solve for x:
    • From `x – 3 = 0`, we get `x = 3`.
    • From `x – 4 = 0`, we get `x = 4`.

The solutions to `x^2 – 7x + 12 = 0` are `x = 3` and `x = 4`.

Checking Solutions

It’s crucial to verify your solutions by substituting them back into the original equation. If the equation holds true, your solutions are correct.

  • Check for x = 3: `(3)^2 – 7(3) + 12 = 9 – 21 + 12 = 0`. Since `0 = 0`, `x = 3` is a correct solution.
  • Check for x = 4: `(4)^2 – 7(4) + 12 = 16 – 28 + 12 = 0`. Since `0 = 0`, `x = 4` is a correct solution.

Key Points for Quick Revision

  • Always rearrange the quadratic equation to the standard form `ax^2 + bx + c = 0` before factoring.
  • Look for the greatest common factor (GCF) first.
  • Identify if it’s a difference of squares (`a^2 – b^2 = (a-b)(a+b)`).
  • For trinomials, use the “splitting the middle term” technique.
  • The Zero Product Rule is the foundation: if `A * B = 0`, then `A = 0` or `B = 0`.
  • Always substitute your found roots back into the original equation to verify their correctness.

Extra Practice Questions

Try to solve the following quadratic equations by factorization:

  1. `x^2 + 8x + 15 = 0`
  2. `4x^2 – 9 = 0`
  3. `x^2 – 2x = 0`
  4. `2x^2 + 11x + 12 = 0`
  5. `x^2 + 10x + 25 = 0`