Section 13, easy algebra

Linear & Quadratic Equations

Linear and quadratic equations are the friendliest algebra in CAT: word problems become one or two equations, and most quadratic questions need only the sum and product of the roots. They also power the optimisation questions hidden in arithmetic.

How CAT asks it

Linear: word problems with two or three unknowns, sometimes with fewer equations than unknowns so only a combination can be found; test scores with negative marking; whole-number solutions of ax + by = c. Quadratic: expressions in the roots; conditions for real or integer roots; equations that turn quadratic after a substitution; the maximum or minimum of a quadratic.

Concepts you must know

Linear

  • n unknowns usually need n independent equations.
  • With fewer equations than unknowns, CAT asks for a combination (such as x + y + z). Add or subtract multiples of the given equations to build it.
  • ax + by = c in whole numbers: find one solution, then step x by b/g and y by a/g in opposite directions, where g = HCF(a, b).
  • For a₁x + b₁y = c₁ and a₂x + b₂y = c₂: one solution if a₁/a₂ ≠ b₁/b₂; none if a₁/a₂ = b₁/b₂ ≠ c₁/c₂; infinitely many if all three ratios are equal.

Quadratic

  • The roots of ax² + bx + c = 0 are
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}
  • Sum of roots = −b/a; product of roots = c/a.
  • Discriminant D = b² − 4ac: D > 0 gives two distinct real roots, D = 0 equal roots, D < 0 no real roots. With rational coefficients, rational roots need D to be a perfect square.
  • The equation with roots α and β is x² − (α + β)x + αβ = 0.
  • α² + β² = (α + β)² − 2αβ; (α − β)² = (α + β)² − 4αβ; α³ + β³ = (α + β)³ − 3αβ(α + β); 1/α + 1/β = (α + β)/αβ.
  • Signs: product < 0 means one positive and one negative root; sum > 0 and product > 0 means both positive.
  • ax² + bx + c is smallest at x = −b/2a when a > 0, and largest there when a < 0.
  • Substitutions such as t = 2ˣ, t = x² or t = x + 1/x turn harder equations into quadratics.

Thought process

  1. Name the unknowns and write one equation per fact.
  2. Before solving, check what is asked: often a combination or an expression in the roots, not the roots themselves.
  3. For quadratics, use sum and product first; find the actual roots only if you must.
  4. Check the domain: counts must be whole and positive, and roots must be real when the story needs them.

Patterns CAT repeats

Numbered so your error log can point at them: “Pattern 2”.

  1. 1Word problems → a linear system.
  2. 2Fewer equations than unknowns: the asked quantity is a combination.
  3. 3Whole-number solutions: ways to pay an amount with two kinds of coins.
  4. 4Scoring with negative marks.
  5. 5Expressions in the roots via sum and product.
  6. 6Conditions on roots: real, equal, integer, opposite signs, one root k more than the other.
  7. 7Equations that become quadratic after a substitution.
  8. 8Max/min of a quadratic, often hidden in a revenue or profit story (Section 6, Example 9).

Worked examples

Attempt each one before opening its solution, then compare methods.

  1. Example 1, combination

    3 pens, 2 pencils and 1 eraser cost ₹52; 4 pens, 3 pencils and 2 erasers cost ₹75. What do one pen, one pencil and one eraser cost together?

    Show solution

    Subtract the first equation from the second: p + q + r = 75 − 52. Answer: ₹23.

  2. Example 2, scoring

    A test gives +4 for a correct answer and −1 for a wrong one. Asha attempts 60 questions and scores 160. How many did she get right?

    Show solution

    c + w = 60 and 4c − w = 160. Adding, 5c = 220. Answer: 44.

  3. Example 3, whole-number solutions

    In how many ways can ₹100 be paid with ₹5 and ₹2 coins, using at least one of each?

    Show solution

    5x + 2y = 100 needs x even: x = 2, 4, …, 18 (x = 20 would leave y = 0). Answer: 9 ways.

  4. Example 4, ages

    A mother’s and daughter’s ages add to 50. Five years ago the mother was 7 times as old as the daughter. Find their present ages.

    Show solution

    M − 5 = 7(D − 5) and M = 50 − D, so 45 − D = 7D − 35 and D = 10. Answer: 40 and 10.

  5. Example 5, consistency

    For what k do 2x + 3y = 7 and 4x + ky = 14 have infinitely many solutions?

    Show solution

    Need 2/4 = 3/k = 7/14. Answer: k = 6.

  6. Example 6, expression in roots

    α and β are the roots of x² − 7x + 10 = 0. Find α² + β².

    Show solution

    (α + β)² − 2αβ = 49 − 20. Check: the roots are 2 and 5, and 4 + 25 = 29. Answer: 29.

  7. Example 7, root condition

    One root of x² − 11x + k = 0 is 3 more than the other. Find k.

    Show solution

    Roots r and r + 3 add to 11, so r = 4 and the roots are 4 and 7. k = product. Answer: 28.

  8. Example 8, counting real-root cases

    For how many integers k with |k| ≤ 10 does x² + kx + 16 = 0 have real roots?

    Show solution

    Need k² − 64 ≥ 0, so |k| ≥ 8: k = ±8, ±9, ±10. Answer: 6.

  9. Example 9, substitution

    Solve 4ˣ − 6 × 2ˣ + 8 = 0.

    Show solution

    With t = 2ˣ, t² − 6t + 8 = 0, so t = 2 or 4. Answer: x = 1 or x = 2.

  10. Example 10, minimum

    Find the minimum value of x² − 6x + 14.

    Show solution

    x² − 6x + 14 = (x − 3)² + 5, which is smallest at x = 3. Answer: 5.

Traps

  • Solving for every variable when only a combination is asked; it wastes time and is sometimes impossible.
  • Sum of roots is −b/a, not b/a.
  • Ignoring the domain: negative people, fractional coins, roots that must be real.

Practice set

Timed, about 2.5 minutes per question. Log every miss in the error log.

  1. 1

    Find the equation whose roots are the reciprocals of the roots of 2x² − 5x + 3 = 0.

    Show answer

    3x² − 5x + 2 = 0: replace x by 1/x and clear the fractions.

  2. 2

    α and β are the roots of x² − 5x + 3 = 0. Find 1/α + 1/β.

    Show answer

    5/3: sum ÷ product.

  3. 3

    5 chairs and 3 tables cost ₹2,900; 3 chairs and 5 tables cost ₹3,500. What do one chair and one table cost together?

    Show answer

    ₹800: adding gives 8 chairs + 8 tables = 6,400.

  4. 4

    How many non-negative whole-number solutions does 3x + 5y = 60 have?

    Show answer

    5: x must be a multiple of 5 from 0 to 20.

  5. 5

    For what k does x² − 8x + k = 0 have equal roots?

    Show answer

    16: D = 64 − 4k = 0.

  6. 6

    Solve 9ˣ − 4 × 3ˣ + 3 = 0.

    Show answer

    x = 0 or 1: with t = 3ˣ, t = 1 or 3.

Log a missFind this chapter’s real CAT questions