Basic Concepts

Quadratic Equation

f(x) = a₀ + a₁x¹ + a₂x² + … a_nxⁿ, where a₁, a₂, ……… a_n are constants, is called a polynomial.

If a₁ a_i Î R where i = 0, 1, 2, ……, n, then we call it a polynomial of a real variable with real coefficients.

If a_n ¹ 0 then we say degree of the polynomial is n. In case of complex coefficients it is called a complex polynomial.

We shall discuss more about the properties of the polynomials. At present, let us look at a particular case i.e. n = 2. When n = 2, then we call it a quadratic polynomial.

        f(x) = a₀ + a₁ + a₂x² = 0 is called quadratic equation.

The general form of the quadratic equation is represented by

                ax² + bx + c = 0.

where a, b, c Î R and a ¹ 0. If all the three coefficients are non-zero then

ax² + bx + c = 0 is called the complete form of the quadratic equation.

Now, ax² + bx + c = 0 can be written as

[If a < 0 we can multiply both sides of the equation by (–1) to make ‘a’ positive].

         = 0

taking square root on both sides

                Þ

                        = 

                Þ x = 

so finally we have two solution (Let us denote them by a and b).

        a =  and b = 

These two solutions of a quadratic equation are called the roots of the equation.

Sum and product of the roots

Let ax² + bx + c = 0 be a quadratic equation whose roots are a and b then

        a + b = 

                = 

Hence, Sum of roots = 

Next, product of roots i.e. a.b

        = 

        = 

product of roots = .

Note:

 Let a and b are the given roots. Then (x – a)(x – b) = 0 must be a equation of 2^nd degree in x with a and b as solutions(roots)

        x² – (a + b) x + ab = 0

        x² – (sum of the roots)x + product of the roots = 0

If you multiply the above equation by any constant (¹ 0), the resultant equation will also be quadratic in nature and have the same roots as parent equation. You may know that, in general for a polynomial equation of degree. i.e.

        a₀ xⁿ + a₁x^n–1 + …… + a_n = 0, a₀ ¹ 0                       …… (i)

If the roots are a₁, a₂ …… + a_n, the polynomial equation can be written as

        a₀ (x – a) ……… (x – a_n) = 0

        a₀ [x_n – S₁x^n–1 + S₂x^n–2 …… + (–1)^r S_r x^n–r +………] = 0        …… (ii)

Where Sr denotes sum of the products of roots taken r at a time eg.

S₃ = a₁a₂a₃ + a₂a₃a₁ + a₃a₁a₂ …… Comparing the coefficients in (i) and (ii) of different powers of x, we observe that

        S₁ = a₁/a₀, S₂ = a₂/a₀ and in general S^r = (–1)^r a_r/a₀

Illustration:

        Let us take an example of a cubic polynomial equation for verifying the      above deductions.

        a₀x³ + a₁x² + a₂x + a₃ = 0, a₀ ¹ 0                                          … (i)

 Solution:

        If a₁, a₂ and a₃ are its three roots then above equations can be written        as

                a₀ (x – a₁)(x – a₂)(x – a₃) = 0

        Þ     a₀ [x³–a₁+a₂+a₃)x² + (a₁a₂+a₂a₃+a₃a₁)x – a₁a₂a₃] = 0        … (ii)

        from (i) and (ii) comparing coefficients of different powers we get

                a₁ + a₂ + a₃ =  = S₁

                a₁a₂ + a₂a₃ + a₃a₁ =  = S₂

        and   a₁a₂a₃ =  = S₃

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