Linear & Quadratic Equations
Scenario
Finding the exact break-even point for a cloud kitchen: where your fixed rent and variable food costs finally match your Swiggy revenue.
Mental model
Equations are balanced scales weighing unknowns.
Linear equations are straight lines indicating constant trade-offs (e.g., spending budget on laptops vs mice). Quadratics are curves (parabolas) showing acceleration or finding a peak/trough (e.g., throwing a ball in the air).
Formula
Linear (one variable): ax + b = 0 -> x = -b/a. Linear (two variables): solve by substitution or elimination. Quadratic: ax^2 + bx + c = 0. Roots: x = (-b +/- sqrt(b^2 - 4ac)) / 2a. Sum of roots = -b/a. Product of roots = c/a. Discriminant D = b^2 - 4ac: D > 0 means 2 real roots, D = 0 means 1 repeated root, D < 0 means no real roots.
For simultaneous linear equations, elimination is fastest when one variable has the same coefficient. For quadratics, always try to factor mentally using sum (-b/a) and product (c/a) before using the quadratic formula.
Worked example
Problem: Solve simultaneously: 2x + 3y = 16 and 4x - y = 6.
- Set up for elimination: Multiply equation 2 by 3 to match the y coefficient: 12x - 3y = 18.
- Add both equations: (2x + 3y) + (12x - 3y) = 16 + 18 -> 14x = 34 -> x = 34/14 = 17/7.
- Wait - use substitution instead for cleaner numbers. From equation 2: y = 4x - 6. Substitute into equation 1: 2x + 3(4x - 6) = 16 -> 2x + 12x - 18 = 16 -> 14x = 34 -> x = 17/7.
- Hmm, messy. Let us try multiplying equation 2 by 3: 12x - 3y = 18. Add to eq 1: 14x = 34. x = 17/7. y = 4(17/7) - 6 = 68/7 - 42/7 = 26/7.
- Sanity-check: Plug back into eq 1: 2(17/7) + 3(26/7) = 34/7 + 78/7 = 112/7 = 16. Correct.
Answer: x = 17/7, y = 26/7.
Problem: Find the roots of x^2 - 7x + 12 = 0.
- Identify: a = 1, b = -7, c = 12. Sum of roots = -(-7)/1 = 7. Product of roots = 12/1 = 12.
- Find two numbers that add to 7 and multiply to 12: 3 and 4 (3+4=7, 3x4=12). Done.
- Roots are x = 3 and x = 4.
- Sanity-check: Discriminant = (-7)^2 - 4(1)(12) = 49 - 48 = 1 > 0. Two real roots confirmed.
Answer: x = 3 and x = 4.
Shortcuts
- Sum and Product of Roots: For ax^2 + bx + c = 0, Sum of roots = -b/a, Product of roots = c/a. (For x^2 - 5x + 6 = 0, sum is 5, product is 6. The numbers 2 and 3 fit this perfectly. Roots found instantly.)
- Option Elimination for Linears: Do not solve simultaneous equations manually if options are given. Just plug the (x, y) options into the simpler equation. (If x + y = 10 and 3x - 2y = 5, and Option A is (5, 5): 5+5=10 checks out, but 15-10=5 also checks out. So A is correct.)
Common mistakes
- Forgetting the negative root of a square: If x^2 = 16, x can be 4 OR -4. Many students only write x = 4 and lose half the solution.
- Dividing by variables: If x^2 = 3x, do NOT divide by x to get x = 3. You lose the x = 0 root. Always factor: x(x - 3) = 0, giving x = 0 or x = 3.
Glossary
- simultaneous
- Things happening or being solved at the exact same time.
- coefficient
- A number placed before a variable to multiply it, like the 3 in 3x.
- substitution
- Replacing one thing with another equivalent thing.
Recall questions
- What is the formula for the sum of roots in a quadratic equation ax^2 + bx + c = 0?
- If the discriminant is negative, what type of roots does the equation have?
- Can a linear equation have two different variables?
Questions & answers
Find the roots of the equation x^2 - 9x + 20 = 0
4, 5
Approach: Sum = 9, Product = 20. The numbers 4 and 5 satisfy this. Hence roots are 4 and 5.
5 tables and 3 chairs cost Rs5500. 3 tables and 2 chairs cost Rs3400. What is the cost of 1 table?
800
Approach: 5t + 3c = 5500. 3t + 2c = 3400. Multiply second by 1.5: 4.5t + 3c = 5100. Subtract from first: 0.5t = 400. So t = 800.