Algebra

Algebra

Concept

Linear Equations — One Solution, Always

A linear equation in one variable (ax + b = 0) always has exactly one solution: x = –b/a. The graph is a straight line crossing the x-axis at that point.

With TWO linear equations in two variables, you have three possibilities:

• One solution — lines intersect at one point
• No solution — lines are parallel (same slope, different intercept)
• Infinite solutions — lines are actually the same line

To solve, use either substitution (solve one for a variable, plug into the other) or elimination (multiply equations so one variable cancels when added).

Quadratic Equations — The Discriminant is Your Friend

The discriminant D = b² – 4ac tells you everything about the roots before you find them:

• D > 0: Two distinct real roots
• D = 0: Two equal real roots (repeated root)
• D < 0: No real roots (complex roots exist, but not in CUET scope)

The quadratic formula gives you the roots directly. But many CUET questions let you factor the quadratic instead — much faster when it works! If x² – 5x + 6 = 0, think: “what two numbers multiply to +6 and add to –5?” → –2 and –3. So (x–2)(x–3) = 0, giving x = 2 or 3.

Vieta’s Formulas — Sum and Product of Roots

For ax² + bx + c = 0 with roots α and β:

• α + β = –b/a (sum of roots)
• αβ = c/a (product of roots)

This is HUGE for problems that ask for expressions involving roots without finding the roots themselves.

Progressions — Patterns That Behave Predictably

An Arithmetic Progression (AP) has a constant difference d between consecutive terms:

• aₙ = a + (n–1)d
• Sum of n terms: Sₙ = n/2[2a + (n–1)d] = n(a + l)/2 where l is the last term

A Geometric Progression (GP) has a constant ratio r:

• aₙ = ar^(n–1)
• Sum of n terms: Sₙ = a(r^n – 1)/(r – 1) when r ≠ 1

Key Formulas

Formula	Use
x = –b/a	Solving linear equation ax + b = 0
x = [–b ± √(b²–4ac)]/2a	Quadratic formula
D = b² – 4ac	Discriminant
α + β = –b/a	Sum of quadratic roots
αβ = c/a	Product of quadratic roots
aₙ = a + (n–1)d	nth term of AP
Sₙ = n/2[2a + (n–1)d]	Sum of n terms of AP
aₙ = ar^(n–1)	nth term of GP
Sₙ = a(r^n – 1)/(r – 1)	Sum of n terms of GP (r ≠ 1)

Worked Example

Q: If α and β are roots of x² – 7x + 10 = 0, find α³ + β³.

Step 1: From Vieta: α + β = 7, αβ = 10 Step 2: Use identity: α³ + β³ = (α + β)³ – 3αβ(α + β) Step 3: = 7³ – 3(10)(7) = 343 – 210 = 133

Answer: 133

Common Errors

• Forgetting to divide by a in quadratic formula → It’s 2a in the denominator, not just 2!
• Sign errors in Vieta’s formulas → α + β = –b/a, αβ = c/a — watch the negative sign for sum
• Using wrong GP sum formula for r < 1 → For r < 1, Sₙ = a(1 – r^n)/(1 – r) is equivalent — same thing
• Confusing AP and GP → AP: add d each time; GP: multiply by r each time