Quadratic Equations for Math Olympiad: Complete Guide

Why This Matters

Quadratic equations are where algebra gets really exciting for Olympiad aspirants. The interplay between factoring, the quadratic formula, discriminant analysis, and the nature of roots creates rich problem-solving opportunities.

For Class 10 students, competition problems test your ability to solve quadratics quickly, analyze discriminants for root behavior, and construct quadratics from given conditions.

Best Strategy

Common Pitfalls

Mistakes:

• Forgetting $a \neq 0$ — A quadratic must have $a \neq 0$.
• Discriminant sign — $D = b^2 - 4ac$. Do not forget the negative sign before $4ac$.
• Completing the square — Add AND subtract the same value to maintain equality.
• Extraneous roots — When equations are derived from word problems, check that roots satisfy the original constraints.

Practice Questions

Try these!

Question 1

Find the nature of roots of $2x^2 - 3x + 5 = 0$.

Solution: $D = (-3)^2 - 4(2)(5) = 9 - 40 = -31 < 0$. No real roots.

Question 2

Find a quadratic equation whose roots are $2+\sqrt{3}$ and $2-\sqrt{3}$.

Solution: Sum = 4, Product = $(2)^2 - (\sqrt{3})^2 = 1$.
Equation: $x^2 - 4x + 1 = 0$.

Question 3

For what value of $k$ does $kx^2 + 4x + 1 = 0$ have equal roots?

Solution: $D = 16 - 4k = 0$, so $k = 4$.

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