2 questions from actual AFCAT papers · all shown free · click option to reveal solution
QUADRATIC EQUATIONS — COMPLETE GUIDE FOR SSC CGL ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
CORE CONCEPT ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
A quadratic equation is any equation where the highest power of x is 2. It always looks like this: ax² + bx + c = 0
Here, a, b, c are numbers (called coefficients), and a cannot be zero. If a = 0, it becomes a simple linear equation — not quadratic. Every quadratic equation has exactly TWO roots (solutions). These roots tell you the values of x that satisfy the equation. The roots are often written as α (alpha) and β (beta).
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ These two formulas are the backbone of almost all SSC questions: Sum of roots: α + β = -b/a Product of roots: α × β = c/a Nature of roots depends on the Discriminant (D): D = b² - 4ac If D > 0 → Two distinct real roots If D = 0 → Two equal (same) real roots If D < 0 → No real roots (imaginary roots) ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
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Quadratic Formula (use when factoring is hard):
Building equation from roots:
Key identity shortcuts:
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If roots are -α and -β, replace x with -x in original equation
When roots are equal, D = 0, so b² = 4ac. Use this to find unknown k in seconds without solving the full equation. ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
Identify a=2, b=-5, c=3
Sum of roots = α + β = -(-5)/2 = 5/2
Product of roots = αβ = 3/2
Use identity → α² + β² = (α + β)² - 2αβ = (5/2)² - 2(3/2) = 25/4 - 3 = 25/4 - 12/4 = 13/4 Answer: 13/4 ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ WORKED EXAMPLE 2 (Forming New Equation) ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ Q: Roots of x² - 6x + 8 = 0 are α and β. Form equation with roots 1/α and 1/β.
a=1, b=-6, c=8
New sum = 1/α + 1/β = (α + β)/αβ = 6/8 = 3/4
New product = 1/α × 1/β = 1/αβ = 1/8
New equation = x² - (3/4)x + 1/8 = 0 Multiply throughout by 8 → 8x² - 6x + 1 = 0 SHORTCUT CHECK: Just swap a and c → 8x² - 6x + 1 = 0. Same answer! Trick works perfectly. ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ #1 MOST COMMON TRAP — STUDENTS LOSE MARKS HERE ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ When finding sum of roots, the formula is -b/a, NOT b/a. For 2x² - 5x + 3 = 0, b = -5. So sum = -(-5)/2 = +5/2. Students forget the negative sign and write 5/2 for b directly — they get -5/2 instead. This one sign error changes the entire answer. Always re-check the sign of b before substituting.
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