NDA Quadratic Equations: Roots, Discriminant and Coefficient Relations

Owned by UPSC NDA Mathematics syllabus. Exact NDA-relevant scope: Quadratic equations with real coefficients within the official Algebra syllabus, including solution methods and root-coefficient relationships needed for NDA-style objective problems.

Quadratic equations with real coefficients within the official Algebra syllabus, including solution methods and root-coefficient relationships needed for NDA-style objective problems.

Factorisation, linear equations, squares / roots and algebraic identities.

• Quadratic equation ax^2+bx+c=0 with a!=0
• Discriminant D=b^2-4ac
• Roots / zeros

• x=[-b +/- sqrt(D)]/(2a)
• D>0: two distinct real roots; D=0: equal real roots; D<0: non-real conjugate roots for real coefficients
• alpha+beta=-b/a
• alpha beta=c/a
• Equation with roots alpha,beta: x^2-(alpha+beta)x+alpha beta=0 up to a non-zero scale factor

• a must be non-zero.
• Discriminant classification assumes real coefficients.
• Root-sum / product relations use the same coefficients and equation scale consistently.
• Squaring both sides in related algebra can introduce extraneous roots, so substitute back when relevant.

1. x^2-5x+6=0 factors to roots 2 and 3; sum 5 and product 6 match coefficient relations.
2. For 2x^2+4x+5=0, D=16-40=-24, so no real roots.
3. If roots have sum 7 and product 10, the monic equation is x^2-7x+10=0.

• Forgetting 2a in the denominator.
• Sign error in -b/a.
• Calling D=0 'no roots' rather than repeated real root.
• Factoring incorrectly and not checking.
• Using root relations on a linear equation after accidental cancellation.

• Knowledge Gap: The concept, definition or theorem needed to solve the item is not understood.
• Recall Gap: The concept was learned but a formula, condition or relation cannot be retrieved accurately.
• Execution Error: The correct method is known but algebra, arithmetic, sign, substitution, diagram or step execution fails.
• Decision / Selection Error: The candidate chooses an inefficient method, continues a time-heavy path, or attempts a low-confidence item without adequate basis.
• Needs Review: Evidence is insufficient or contradictory; do not mislabel the error until another worked item is checked.
• Time pressure note: Time pressure is treated as a contributing condition, not automatically as the root cause.

Try factorisation when coefficients are simple. Use discriminant first if the question asks about nature of roots. Use sum / product relations for symmetric root expressions instead of solving individual roots. Keep the quadratic formula as the general fallback.

Negative marking is a real exam condition, but this page does not promise a universal attempt threshold. Use method certainty and option evidence before committing to a low-confidence answer.

• Required reviewer: NDA Mathematics faculty with demonstrable subject competence.