➗ Equations (Linear & Quadratic)

Day 1 | ⏱️ Part of 4-Hour Quant Block | TCS Interview Prep

📌 Subtopics

• Linear Equations: Single variable, two variables, system of linear equations
• Quadratic Equations: Standard form ax² + bx + c = 0, nature of roots
• Discriminant (b² - 4ac)
• Sum and product of roots
• Word problems leading to equations

📖 Comprehensive Notes

1. Linear Equations

A linear equation is an algebraic equation where each term has an exponent of at most $1$.

• Single Variable: $ax + b = 0 \implies x = -\frac{b}{a}$.
• Two Variables (System of Equations): $a_1 x + b_1 y = c_1$ $a_2 x + b_2 y = c_2$
  • Solve using Substitution or Elimination methods.
  • Consistency Conditions:
    1. Unique Solution: $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$ (Intersecting lines).
    2. Infinite Solutions: $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ (Coincident lines).
    3. No Solution: $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$ (Parallel lines).

Example: Solve the system: $2x + 3y = 12$ and $x - y = 1$. From the second equation: $x = y + 1$. Substitute into the first: $2(y+1) + 3y = 12 \implies 5y + 2 = 12 \implies 5y = 10 \implies y = 2$. Hence, $x = 3$.

2. Quadratic Equations

A quadratic equation is of the standard form $ax^2 + bx + c = 0$ (where $a \neq 0$).

• Quadratic Formula: The roots are given by: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

3. Discriminant ($D = b^2 - 4ac$)

The discriminant determines the nature of the roots:

• If $D > 0$: Roots are real and distinct.
• If $D = 0$: Roots are real and equal ($x = -\frac{b}{2a}$).
• If $D < 0$: Roots are complex/conjugate imaginary.
• If $D$ is a perfect square, roots are rational (assuming $a, b, c$ are rational).

Example: Find the nature of roots of $2x^2 - 5x + 3 = 0$. $D = (-5)^2 - 4(2)(3) = 25 - 24 = 1$. Since $D > 0$ and is a perfect square, the roots are real, distinct, and rational.

4. Relation Between Roots and Coefficients

If $\alpha$ and $\beta$ are the roots of $ax^2 + bx + c = 0$:

• Sum of Roots: $\alpha + \beta = -\frac{b}{a}$
• Product of Roots: $\alpha \cdot \beta = \frac{c}{a}$
• A quadratic equation can be reconstructed as: $x^2 - (\text{Sum of Roots})x + (\text{Product of Roots}) = 0$

Example: Find the quadratic equation whose roots are $3$ and $-4$. $\text{Sum} = 3 + (-4) = -1$ $\text{Product} = 3 \times (-4) = -12$ Equation: $x^2 - (-1)x + (-12) = 0 \implies x^2 + x - 12 = 0$.

5. Word Problems Leading to Equations

Translate real-world conditions into equations by identifying:

• The unknown quantity (assign as $x$).
• Equality key indicators: “is”, “becomes”, “equals”, “results in”.

Example: The sum of a number and its reciprocal is $\frac{10}{3}$. Find the number. Let the number be $x$. Equation: $x + \frac{1}{x} = \frac{10}{3} \implies \frac{x^2 + 1}{x} = \frac{10}{3} \implies 3x^2 - 10x + 3 = 0$. Factoring: $(3x - 1)(x - 3) = 0 \implies x = 3$ or $x = 1/3$.

🌐 Learning Resources

✅ Practice Links

📝 Key Formulas

Linear:     ax + b = 0  →  x = -b/a
Quadratic:  ax² + bx + c = 0  →  x = (-b ± √(b²-4ac)) / 2a

Sum of roots  = -b/a
Product of roots = c/a

Discriminant D = b² - 4ac
  D > 0 → Two distinct real roots
  D = 0 → Two equal real roots
  D < 0 → No real roots (complex)

🔗 Related Topics

• Ages — Age problems are solved by forming linear equations
• Number System HCF LCM — Number properties used in equation solving
• Profit and Loss — Word problems often reduce to equations
• Time and Work — Work rate problems use equation setup
• Series and Progressions — AP/GP formulas involve equation solving