Quadratic equations Assignment Help

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Quadratic equations:

An equation of the form anxn + an - 1xn - 1 + an - 2xn - 2 + .... + a1x + a0 = 0 (a0, a1...., an are real coefficients and an ≠ 0) is known as a polynomial equation of degree n. That equation is known as a linear equation if n=1, then quadratic equation if n=2, then cubic equation if n=3, then bi-quadratic equation if n=4...and so on.

Definition:

An equation shows as ax2 + bx + c = 0, where a ≠ 0 and a, b, c are real numbers, is known as a quadratic equation. The numbers a, b, c defined as the coefficients of the quadratic equation.

Root of a quadratic equation:

A root of the quadratic equation is a number a (complex or real) so that aα2 + bα + c = 0. The roots of the quadratic equation are provided by x = 2241_Quadratic equations.png. Let β and α be two roots of the provided quadratic equation then α + β = -b/a and αβ = c/a.

Discriminant of a quadratic equation:

The quantity D (D= b2 - 4ac) is known as the discriminant of a quadratic equation.

  •   The quadratic equation has same and real roots if and only if D = 0

      i.e. b2 - 4ac = 0.

  •   The quadratic equation has distinct and real roots if and only if D > 0

      i.e. b2 - 4ac > 0

  •   The quadratic equation has complex roots with non-zero imaginary parts if and only if D < 0 i.e. b2 - 4ac < 0.  If p + iq (p and q being real) is a root of the quadratic equation where i = √-1, then p - iq is also a root of the quadratic equation.

 

Problem: If the roots of the quadratic relation x2 - ax + b = 0 are differ and real by a quantity less than 1, then show that 2192_Quadratic equations1.png.

Key concept:  Since the roots of the provided equation  differ by less then 1 therefore roots are distinct and real, therefore the D>0. Also the difference between the roots could be less then one. Apply the identity  (α - β)2 = (α + β)2 - 4αβ

Solution:                   We know that D=a2 - 4b > 0 => 310_Quadratic equations2.png.................(1)

            Now  (α - β)2 = (α + β)2 - 4αβ = a2 - 4b , where a and b are roots of the provided equation.

            =>|α - β|2 = a2 - 4b, but provided that |α - β| < 1 => |a2 - 4b| < 1

                        => a2 - 4b < 1 => a2-1/4 < b                          ......(2)

                        From (1) and (2),  2443_Quadratic equations3.png

 

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