**Discriminant Laws of any Equation**:The nature of the** roots of an equation** depend on the value of the expression** b² -4ac,** which is called its **discriminant.**

**Discriminant Law one: 1**

**If b² -4ac =0** then the roots will be **– b/ 2a and – b / 2a**

So the roots are real and repeated equal.

**Discriminant Law two: 2**

**If the roots b²- 4ac < 0** then **√ b² – 4ac** will be **imaginary.**

So the roots are **complex** / **imaginary and distinct**/**unequal.**

**Discriminant Law three: 3**

**If the roots b²- 4ac > 0 **then **√ b² – 4ac **will be** real.**

However, If **b² – 4ac** is a perfect square then **√ b² – 4ac **will be rational, and irrational

** So the roots are rational**, otherwise** irrational.**

**Examples:**

**Discriminant Laws of any Equation.**

We discuss the Discriminant Laws of any expression.

**(1) x² + 2x + 3 = 0 (2) 2 x² + 5x -1 =0**

**(3) x² -7x + 3 = 0 (4) 9 x² -12x +4 =0**

**Solution:**

**(1) x² + 2x + 3 = 0 **

Comparing** x² + 2x + 3 = 0 with Ax ² +bx +c= 0,** we have

**a =1, b = 2, c = 3**

**Disc = b² – 4ac **

**Disc = (2)² – 4(1)(3)= -8 < 0**

**Disc < 0**

See discriminant Law two (2)

**therefore, the roots are complex /imaginary/unequal.**

** (2) 2 x² + 5x -1 =0**

Comparing 2** x² + 5x -1 = 0 with Ax ² +bx +c= 0,** we have

**a =2, b = 5, c =-1**

**Disc = b² – 4ac **

**Disc = (5)² – 4(2)(-1)=25+ 8 =33**

**Disc > 0 (means discriminant greater than zero)**

See discriminant Law Three (3)

**So the roots are irrational and unequal.**

**Solution :**

** (3) x² -7x + 3 = 0**

Comparing** 2 x² -7x +3 = 0 with Ax ² +bx +c= 0,** we have

**a =2, b = -7, c = 3**

**Disc = b² – 4ac **

**Disc = (-7)² – 4(2)(3)=49-24=25=5²**

**Disc >0**

**Therefore, the roots are real and unequal.**

**Solution:**

** (4) 9 x² -12x +4 =0**

Comparing** 9x² -12x + 4 = 0 with Ax ² +bx +c= 0,** we have

**a =9, b =-12, c = 4**

**Disc = b² – 4ac **

**Disc = (-12)² – 4(9)(4)= 144-144 = 0**

**Disc = 0**

See discriminant** Law one (1)**

**Therefore, the roots are real and equal.**

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