Solution by factorisation — Class 10 CBSE Mathematics MCQs with Solutions
Free Class 10 CBSE Mathematics Solution by factorisation MCQs with step-by-step solutions (4 questions). Part of Quadratic Equations. Practise online on Prepizo — no login needed.
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Questions with solutions
Q1 — Solution by factorisation · easy · theory
Which statement correctly identifies why $x^2 + x + 1 = 0$ cannot be solved by factorisation over the real numbers?
A. Because the coefficient of $x^2$ is 1
B. Because it has no real roots (discriminant is negative) ✓ Correct
C. Because it has equal roots
D. Because there is no constant term
Solution: Factorisation works when the quadratic can be written as $(px+q)(rx+s) = 0$, which happens only when the roots are real. Here, $\Delta = 1^2 - 4(1)(1) = -3 < 0$, so the roots are complex. Without real roots, factorisation into real linear factors is impossible.
Q2 — Solution by factorisation · medium · theory
Why does the equation $(2x+1)(x-3) = 0$ lead to two roots via the zero-product property, and why is this related to factorisation?
A. Because we set each factor to zero and solve two linear equations separately ✓ Correct
B. Because multiplying the factors gives the roots directly
C. Because the product of the factors must equal zero
D. Because the coefficients are small integers
Solution: Factorisation reduces a quadratic to the product form $(px+q)(rx+s) = 0$. By the zero-product property, this is satisfied only when $px+q=0$ or $rx+s=0$, giving us two linear equations to solve. Here, $2x+1=0 \Rightarrow x=-\frac{1}{2}$ and $x-3=0 \Rightarrow x=3$. This is the conceptual basis for why factorisation works.
Q3 — Solution by factorisation · medium · theory
For the quadratic $3x^2 - 12x + 9 = 0$, why is factorisation preferable to the quadratic formula in this case?
A. Because all coefficients are divisible by 3, simplifying to $x^2 - 4x + 3 = 0$, which factors easily as $(x-1)(x-3) = 0$ ✓ Correct
B. Because the quadratic formula only works for $a=1$
C. Because the discriminant is zero
D. Because the roots must be integers
Solution: Factorisation is strategic. Here, we can factor out 3: $3(x^2 - 4x + 3) = 0$, simplifying to $x^2 - 4x + 3 = 0$, which factors as $(x-1)(x-3)=0$ by inspection. This avoids fractions and is faster than applying the quadratic formula. The choice between factorisation and formula depends on recognizing factorable patterns.
Q4 — Solution by factorisation · hard · theory
Consider the quadratic $(x - 2)^2 = 0$, which simplifies to $x^2 - 4x + 4 = 0$. Why is this a perfect-square quadratic, and what does it reveal about the relationship between the discriminant and the quadratic's factorisation?
A. It is a perfect square because it factors as $(x-2)^2$, a repeated linear factor. The discriminant is zero, confirming a double root at $x=2$. ✓ Correct
B. It is a perfect square because the constant term equals the square of half the $x$-coefficient; it has two distinct roots.
C. Perfect squares always have positive discriminants and complex roots.
D. The relationship between discriminant and factorisation does not apply to perfect squares.
Solution: A perfect-square quadratic $(x-k)^2 = x^2 - 2kx + k^2$ has the form where the discriminant $\Delta = (-2k)^2 - 4(1)(k^2) = 4k^2 - 4k^2 = 0$. This zero discriminant corresponds to the repeated root $x = k$ with multiplicity 2. Perfect squares are the unique case where a quadratic factors into an identical linear factor twice.