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**Solving a quadratic equation**

An equation of the form ax^{2} + 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, b^{2} - 4ac is called the discriminant of the quadratic equation and is denoted by the capital letter 'D'.

Discriminant (D) = b^{2} - 4ac

The roots of the quadratic equation are x = (-b Â± âˆš(b^{2} - 4ac))/2a

If the discriminant is greater than zero, then the roots of a quadratic equation are real and distinct.

b^{2} - 4ac > 0, Roots: x = (-b + âˆš(b^{2} - 4ac))/2a and (-b - âˆš(b^{2} - 4ac))/2a

If the discriminant is zero, then the roots of a quadratic equation are real and equal.

b^{2} - 4ac = 0, Roots:x = -b/2a and -b/2a.

Ex: Discriminant (D) of 2x^{2} + 2x + 1 = 0

2^{2} - 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 b^{2} - 4ac is negative.

âˆ´ x = (-b Â± âˆš((-1)(4ac-b^{2}))/2a

â‡’ x = (-b Â± iâˆš(4ac - b^{2}))/2a

Therefore, the roots of the quadratic equation, where the discriminant is less than zero can be obtained by using the formulae:

x = (-b + iâˆš(4ac - b^{2}))/2a and (-b - iâˆš(4ac - b^{2}))/2a

The roots of the quadratic equation are complex numbers and they are conjugate to each other.

Ex: Solve the quadratic equation 2x^{2} + 2x + 1 = 0.

Sol:

x = (-b + iâˆš(4ac - b^{2}))/2a and (-b - iâˆš(4ac - b^{2}))/2a

x = (-2 + iâˆš(4x2x1 - 2^{2}))/(2x2) and (-2 - iâˆš(4x2x1 - 2^{2}))/(2x2)

â‡’ x = (-2 + iâˆš(4x2x1 - 2^{2}))/(2x2) and (-2 - iâˆš(4x2x1 - 2^{2}))/(2x2)

â‡’ x = (-2+iâˆš4)/4 and (-2-iâˆš4)/4

âˆ´ 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, ax^{2} + bx + c = 0, is -b/a.

The product of the roots of the quadratic equation, ax^{2} + bx + c = 0, is c/a.

**Solving a quadratic equation**

An equation of the form ax^{2} + 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, b^{2} - 4ac is called the discriminant of the quadratic equation and is denoted by the capital letter 'D'.

Discriminant (D) = b^{2} - 4ac

The roots of the quadratic equation are x = (-b Â± âˆš(b^{2} - 4ac))/2a

If the discriminant is greater than zero, then the roots of a quadratic equation are real and distinct.

b^{2} - 4ac > 0, Roots: x = (-b + âˆš(b^{2} - 4ac))/2a and (-b - âˆš(b^{2} - 4ac))/2a

If the discriminant is zero, then the roots of a quadratic equation are real and equal.

b^{2} - 4ac = 0, Roots:x = -b/2a and -b/2a.

Ex: Discriminant (D) of 2x^{2} + 2x + 1 = 0

2^{2} - 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 b^{2} - 4ac is negative.

âˆ´ x = (-b Â± âˆš((-1)(4ac-b^{2}))/2a

â‡’ x = (-b Â± iâˆš(4ac - b^{2}))/2a

Therefore, the roots of the quadratic equation, where the discriminant is less than zero can be obtained by using the formulae:

x = (-b + iâˆš(4ac - b^{2}))/2a and (-b - iâˆš(4ac - b^{2}))/2a

The roots of the quadratic equation are complex numbers and they are conjugate to each other.

Ex: Solve the quadratic equation 2x^{2} + 2x + 1 = 0.

Sol:

x = (-b + iâˆš(4ac - b^{2}))/2a and (-b - iâˆš(4ac - b^{2}))/2a

x = (-2 + iâˆš(4x2x1 - 2^{2}))/(2x2) and (-2 - iâˆš(4x2x1 - 2^{2}))/(2x2)

â‡’ x = (-2 + iâˆš(4x2x1 - 2^{2}))/(2x2) and (-2 - iâˆš(4x2x1 - 2^{2}))/(2x2)

â‡’ x = (-2+iâˆš4)/4 and (-2-iâˆš4)/4

âˆ´ 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, ax^{2} + bx + c = 0, is -b/a.

The product of the roots of the quadratic equation, ax^{2} + bx + c = 0, is c/a.