Mathematics RU

Exams · Russian state exam, profile level

No. 18 A problem with a parameter

The equation has a letter a in it. Find every a for which the equation has exactly two roots.

How to solve it

Equations with a parameter

An equation with a parameter has a letter besides $x$, usually $a$: $x^2 - 2ax + 9 = 0$. Its value is not given, and you need every value of $a$ for which the equation behaves as asked, for example has exactly two roots. The answer is not a number but a set of values of $a$.

Step by step

  1. See where the parameter sits. If it is in the coefficient of $x^2$, deal separately with the case where that coefficient is zero: the equation becomes linear.
  2. A quadratic equation has two roots exactly when its discriminant is positive. Write $D$ in terms of $a$.
  3. Solve the inequality $D > 0$ for $a$.
  4. Write the answer as intervals of $a$.
The coefficient of $x$. The parameter may sit here. The constant term. The parameter may sit here as well. Example: $x^2 - 2ax + 9 = 0$. Here $D = 4a^2 - 36 > 0$, so $a^2 > 9$: $a < -3$ or $a > 3$. Answer: $(-\infty, -3) \cup (3, +\infty)$.

Common mistakes

  • Solving $D \ge 0$ instead of $D > 0$. At $D = 0$ there is only one root.
  • Getting only $a > 3$ out of $a^2 > 9$ and losing the negative values.
  • Skipping the case where the parameter makes the coefficient of $x^2$ zero.

Example

Mathematics

Chapters

Introduction

  1. 0 What mathematics is

I Numbers

  1. 1 Counting
  2. 2 Zero and minus
  3. 3 Primes
  4. 4 Divisibility
  5. 5 Fractions
  6. 6 Root two

II Algebra

  1. 7 Letters
  2. 8 Equations
  3. 9 Functions and graphs
  4. 10 Quadratic equations
  5. 11 Systems and inequalities
  6. 12 Powers and logarithms
  7. 13 Sequences
  8. 14 Polynomials
  9. 15 Complex numbers

III Geometry

  1. 16 Euclid's Elements
  2. 17 The triangle
  3. 18 Pythagoras' theorem
  4. 19 The circle and π
  5. 20 Area and volume
  6. 21 Trigonometry
  7. 22 Vectors
  8. 23 Symmetry
  9. 24 Conic sections

IV Calculus

  1. 25 Limits
  2. 26 The derivative
  3. 27 Extrema
  4. 28 The integral
  5. 29 The number e
  6. 30 Series
  7. 31 Differential equations
  8. 32 Several variables
  9. 33 Fourier series
  10. 34 Complex analysis

V Linear algebra

  1. 35 Matrices
  2. 36 Gaussian elimination
  3. 37 Vector spaces
  4. 38 Eigenvectors
  5. 39 Least squares and SVD

VI Structures

  1. 40 Groups
  2. 41 Remainders and ciphers
  3. 42 Rings and fields
  4. 43 Galois theory
  5. 44 Zeta and curves

VII Chance & data

  1. 45 Combinatorics
  2. 46 Graphs
  3. 47 Probability
  4. 48 Random variables
  5. 49 Statistics
  6. 50 Markov and information

VIII Foundations

  1. 51 Logic and sets
  2. 52 Infinities
  3. 53 Rigorous analysis
  4. 54 Measure and integral
  5. 55 Functional analysis
  6. 56 Gödel and Turing

IX Horizons

  1. 57 Topology
  2. 58 Non-Euclidean geometries
  3. 59 Chaos and fractals
  4. 60 The frontier