**We have already studied the quadratic formula,**

**The term which is inside the square root symbol is called**

*the discriminant*.**It is used to determine the nature of the roots of a quadratic equation. We can also determine the number of real roots for a quadratic equation with this number. The following table will give us the relation between the discriminant and the nature of the roots.**

Discriminant | Nature of roots | Number of real roots |

b^{2} - 4ac > 0 | Two real roots | 2 |

b^{2} - 4ac = 0 | Double root | 1 |

b^{2} - 4ac < 0 | No real roots | 0 |

**Example 1:**

**Find the discriminant value for the equation**

*x*^{2}+ 5*x*+ 6 = 0 and determine the number of real roots.**Solution:**

**Step 1:**

**Given equation is**

*x*^{2}+ 5*x*+ 6 = 0.**It is in the form**

*ax*^{2}+*bx*+*c*= 0, where*a*= 1,*b*= 5, and*c*= 6.**Step 2:**

**The discriminant =**

*b*^{2}- 4*ac*

*b*^{2}- 4*ac*= (5)^{2}- 4(1)(6) = 25 - 24 = 1 > 0**=> The equation has two real roots.**

**=> The graph of this equation will touch the**

*x*-axis twice.**Step 3:**

**So, the given equation has two real roots.**

**Example: 2**

**Find the discriminant value of**

*x*^{2}- 12*x*+ 36 = 0 and determine the number of real roots.**Solution:**

**Step 1:**

**Given equation is**

*x*^{2}- 12*x*+ 36 = 0.**It is in the form**

*ax*^{2}+*bx*+*c*= 0 where,*a*= 1,*b*= -12, and*c*= 36.**Step 2:**

**Discriminant =**

*b*^{2}- 4*ac*= (-12)^{2}- 4(1)(36) = 144 - 144 = 0**=> The equation has a double root.**

**=> The graph of this equation touches the**

*x-*axis in only one point.**Step 3:**

**So, the given equation has a double root.**

**Example: 3**

**Find the discriminant value 2**

*x*^{2}+*x*+ 3 = 0 and determine the number of real roots.**Solution:**

**Step 1:**

**The given quadratic equation is 2**

*x*^{2}+*x*+ 3 = 0.**It is in the form**

*ax*^{2}+*bx*+*c*= 0 where,*a*= 2,*b*= 1, and*c*= 3.**Step 2:**

*b*^{2}- 4*ac*= (1)^{2}- 4(2)(3) = 1 - 24 = -23 < 0**=> The equation has no real roots.**

**=> The graph of this equation does not touch the**

*x-*axis.**Step 3:**

**So, the given quadratic equation has no real roots.**

**Related Articles:**