**Roots of a Quadratic Equation with Real Coefficients**

An equation of the form ax² + bx + c = 0 … (i)

Where a ≠ 0, a, b, c є R is called quadratic equation with real coefficients.

The quality D = b² – 4ac is known as the discriminant of the quadratic equation in (i) whose roots are given by \(\alpha =\frac{-b+\sqrt{{{b}^{2}}-4ac}}{2a}\) and \(\beta =\frac{-b-\sqrt{{{b}^{2}}-4ac}}{2a}\).

The nature of the roots is as given below:

- The roots are real and distinct if D > 0.
- The roots are real and equal if D = 0
- The roots are complex with non-zero imaginary part if D < 0.
- The roots are rational if a, b, c is rational, and D is a perfect square.
- The roots are of the form p + √q (p, q є Q) if a, b, c is rational, and D is not a perfect square.
- If a = 1, b, c є I and the roots are rational numbers, then these roots must be integers.
- If a quadratic equation in x has more than two roots, then it is an identity in x that is a = b = c = 0.

**Example:** Solve for real x (x² – 1) (x + 2) + 1 = 0

**Solution:** we have,

x (x² – 1) (x + 2) + 1

⇒ x (x – 1) (x + 1) (x + 2) + 1 = 0

⇒ [ x(x+1)] [(x-1) (x+2)] + 1=0

⇒ (x² + x) (x² + x -2) +1 =0

⇒ y(y -2) + 1 = 0 where y = x² + x

⇒ y² -2y +1 = 0

⇒ (y – 1)² = 0

⇒ y= 1

⇒ x² + x = 1

⇒ x² + x – 1 = 0

⇒ \(x=\frac{-1\pm \sqrt{5}}{2}\).

Hence, the roots of the given equations are \(\frac{1-+\sqrt{5}}{2}\) and \(\frac{-1-\sqrt{5}}{2}\).