Notes On Quadratic Equations - CBSE Class 11 Maths
Solving a quadratic equation An equation of the form ax2 + bx + c = 0, where the coefficients a, b and c are real numbers and a is not equal to zero, is called a quadratic equation. In a quadratic equation, b2 - 4ac is called the discriminant of the quadratic equation and is denoted by the capital letter 'D'. Discriminant (D) = b2 - 4ac The roots of the quadratic equation are x = (-b ± √(b2 - 4ac))/2a If the discriminant is greater than zero, then the roots of a quadratic equation are real and distinct. b2 - 4ac > 0, Roots: x = (-b + √(b2 - 4ac))/2a and (-b - √(b2 - 4ac))/2a If the discriminant is zero, then the roots of a quadratic equation are real and equal. b2 - 4ac = 0, Roots:x = -b/2a and -b/2a. Ex: Discriminant (D) of  2x2 + 2x + 1 = 0 22 - 4 x 2 x 1 = 4 - 8 = -4 < 0 The discriminant obtained is -4 , which is less than zero. Since the discriminant is less than zero, the value of b2 - 4ac is negative. ∴ x = (-b ± √((-1)(4ac-b2))/2a ⇒ x = (-b ± i√(4ac - b2))/2a Therefore, the roots of the quadratic equation, where the discriminant is less than zero can be obtained by using the formulae: x = (-b + i√(4ac - b2))/2a and (-b - i√(4ac - b2))/2a The roots of the quadratic equation are complex numbers and they are conjugate to each other. Ex: Solve the quadratic equation 2x2 + 2x + 1 = 0. Sol: x = (-b + i√(4ac - b2))/2a and (-b - i√(4ac - b2))/2a x = (-2 + i√(4x2x1 - 22))/(2x2) and (-2 - i√(4x2x1 - 22))/(2x2) ⇒  x = (-2 + i√(4x2x1 - 22))/(2x2) and (-2 - i√(4x2x1 - 22))/(2x2) ⇒  x = (-2+i√4)/4 and (-2-i√4)/4 ∴ x = (-1+ i)/2 and (-1- i)/2 A quadratic equation has two roots. The fundamental theorem of algebra states that a polynomial equation has at least one root. A polynomial equation of degree n has n roots. The sum of the roots of the quadratic equation, ax2 + bx + c = 0, is -b/a. The product of the roots of the quadratic equation, ax2 + bx + c = 0, is c/a.

#### Summary

Solving a quadratic equation An equation of the form ax2 + bx + c = 0, where the coefficients a, b and c are real numbers and a is not equal to zero, is called a quadratic equation. In a quadratic equation, b2 - 4ac is called the discriminant of the quadratic equation and is denoted by the capital letter 'D'. Discriminant (D) = b2 - 4ac The roots of the quadratic equation are x = (-b ± √(b2 - 4ac))/2a If the discriminant is greater than zero, then the roots of a quadratic equation are real and distinct. b2 - 4ac > 0, Roots: x = (-b + √(b2 - 4ac))/2a and (-b - √(b2 - 4ac))/2a If the discriminant is zero, then the roots of a quadratic equation are real and equal. b2 - 4ac = 0, Roots:x = -b/2a and -b/2a. Ex: Discriminant (D) of  2x2 + 2x + 1 = 0 22 - 4 x 2 x 1 = 4 - 8 = -4 < 0 The discriminant obtained is -4 , which is less than zero. Since the discriminant is less than zero, the value of b2 - 4ac is negative. ∴ x = (-b ± √((-1)(4ac-b2))/2a ⇒ x = (-b ± i√(4ac - b2))/2a Therefore, the roots of the quadratic equation, where the discriminant is less than zero can be obtained by using the formulae: x = (-b + i√(4ac - b2))/2a and (-b - i√(4ac - b2))/2a The roots of the quadratic equation are complex numbers and they are conjugate to each other. Ex: Solve the quadratic equation 2x2 + 2x + 1 = 0. Sol: x = (-b + i√(4ac - b2))/2a and (-b - i√(4ac - b2))/2a x = (-2 + i√(4x2x1 - 22))/(2x2) and (-2 - i√(4x2x1 - 22))/(2x2) ⇒  x = (-2 + i√(4x2x1 - 22))/(2x2) and (-2 - i√(4x2x1 - 22))/(2x2) ⇒  x = (-2+i√4)/4 and (-2-i√4)/4 ∴ x = (-1+ i)/2 and (-1- i)/2 A quadratic equation has two roots. The fundamental theorem of algebra states that a polynomial equation has at least one root. A polynomial equation of degree n has n roots. The sum of the roots of the quadratic equation, ax2 + bx + c = 0, is -b/a. The product of the roots of the quadratic equation, ax2 + bx + c = 0, is c/a.

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