Theory of Equations

Every equation of the nth degree has a total ‘n’ real or imaginary roots. If α is the root of Equation f (x) = 0, then the polynomial f (x) is exactly divisible by (x – α), i.e., (x – α) is the factor of the given polynomial f (x).

In algebra, the study of algebraic equations, which are equations defined by a polynomial, is called the theory of equations. A polynomial is an expression consisting of one or more terms. The main difficulty of the theory of equations was to know when an algebraic equation has an algebraic solution. In this article, we will learn about the theory of equations and examples of solving equations.

The following are some important concepts covered under the theory of equations.

  • Linear equations
  • Simultaneous linear equations
  • Finding the integer solutions of an equation or of a system of equations
  • Systems of polynomial equations

Important Points to Remember

The important concepts in the theory of equations are given below:

  1. The general form of a quadratic equation in x is given by ax2 + bx + c  = 0
  2. The roots are given by x = (-b±√(b2 – 4ac))/2a
  3. If α and β are the roots of the equation ax2 + bx + c  = 0, a ≠ 0, then the sum of roots, α + β = -b/a.

Product of roots,  αβ = c/a

  1. If the sum and product of roots are known, then the quadratic equation is given by x2 – (sum of roots)x + product of roots = 0
  1. For a quadratic equation, b2 – 4ac is known as the discriminant denoted by D.
  2. If D = 0, the equation will have two equal real roots.
  3. If D > 0, then the equation will have two distinct real roots.
  4. If D < 0, then the equation has no real roots. 
  5. The graph of a quadratic equation is a parabola. The parabola will open upwards if a >0, and open downwards if a < 0.
  6. If a > 0, when x = -b/2a, f(x) attains its minimum value.
  7. If a < 0, when x = -b/2a, f(x) attains its maximum value.