Quadratic Equations

What the exam asks

Two equations are given, one in x and one in y. You find the roots and compare: x > y, x < y, x ≥ y, x ≤ y, or x = y / relation cannot be established. Mains also uses quantity comparison (Quantity I vs Quantity II), where a quantity may come from arithmetic instead of an equation. Target: 30 to 40 seconds per question.

Finding roots fast (factorisation)

For ax² + bx + c = 0: find two numbers whose product = a x c and sum = b. Split the middle term, then the roots are those two numbers with signs reversed, divided by a.

Example: 2x² - 11x + 15 = 0. a x c = 30, need sum -11: -5 and -6. Roots = 5/2 and 6/2 = 2.5 and 3.

Sign rule (find signs without solving)

Key formulas

• Roots = [-b ± sqrt(b² - 4ac)] / 2a
• Sum of roots = -b/a; product of roots = c/a
• D = b² - 4ac. D > 0: two real roots; D = 0: equal roots; D < 0: no real roots.
• x² = k gives x = +sqrt(k) and -sqrt(k) - two roots. x = sqrt(k) gives only the positive root.

How to compare

1. Write both roots of x and both roots of y on a number line.
2. If every x lies to the right of every y: x > y. If every x lies to the left: x < y.
3. If they touch at one common value and otherwise x is right: x ≥ y (similarly x ≤ y).
4. If the ranges overlap: relation cannot be established.

Worked example

I. x² - 13x + 42 = 0 and II. y² - 15y + 56 = 0

x: (x - 6)(x - 7) = 0, x = 6, 7. y: (y - 7)(y - 8) = 0, y = 7, 8. The value 7 is common and the remaining x is smaller, so x ≤ y.

Worked example 2 (quantity comparison)

Quantity I: x, where x² = 64. Quantity II: 8.

x = 8 or -8. So Quantity I ≤ Quantity II.

Common traps

• Forgetting the negative root of x² = k.
• Reversing signs incorrectly: if the factors are (x - 5)(x + 3), roots are 5 and -3.
• Comparing only one root of each equation - you must compare every pair.
• Equal roots: (x - 4)² = 0 gives only x = 4.