PREVIOUSLY ASKED IN:
CTET 2026
Answer
\( 4x - 8y + 3 \)
Explanation
Notice that \( 4x^2 + 16y^2 - 16xy = (2x - 4y)^2 \). Let \( z = 2x - 4y \). The expression becomes: \( 5z + 6z^2 - 6 = 6z^2 + 5z - 6 \). Factoring the quadratic expression: \( 6z^2 + 9z - 4z - 6 = 3z(2z + 3) - 2(2z + 3) = (3z - 2)(2z + 3) \). Substituting back \( z = 2x - 4y \): First factor = \( 3(2x - 4y) - 2 = 6x - 12y - 2 \). Second factor = \( 2(2x - 4y) + 3 = 4x - 8y + 3 \). The second factor is present in the options.
Key Points
- > Identifying the hidden perfect square in an algebraic expression is the first step of factorization.
- > The formula \( (a-b)^2 = a^2 - 2ab + b^2 \) is applied here.
- > Substituting a complex repeating term with a single variable (like z) simplifies quadratic equations.
- > In middle-term splitting, the product and sum of the split terms must correctly match the quadratic coefficients.
- > Here, \( 6 \times (-6) = -36 \); the numbers 9 and -4 add up to 5 and multiply to -36.
- > It's mandatory to substitute the original variables back into the factors at the end.
- > Pattern recognition is highly crucial for complex algebraic factorization.
Additional Information
Factorization Techniques
| Method | Formula / Form | When to Use |
|---|---|---|
| Taking Common | ab + ac = a(b+c) | When terms share common factors |
| Middle Term Split | ax² + bx + c | Quadratic expressions |
| Difference of Squares | a² - b² = (a+b)(a-b) | When two squares are subtracted |
| Perfect Square | a² ± 2ab + b² | When expression forms (a±b)² |
Memory Tips
- Substitution Trick: If a large bracketed term repeats, substitute it with a single variable to simplify the problem.
- Sign Check: In quadratics, if the constant term is negative, the split middle terms must have opposite signs.