Quick recap of Quadratic formula

A quadratic equation is an equation of the form ax^{2} + bx + c = 0 where a, b, and c are constants and a ≠ 0

For any such quadratic equation, quadratic formula is:

Roots(x_{1}, x_{2}) = |
−b ± √ b2 − 4ac |

2a |

Discriminant explained

In the quadratic formula above, the quantity **"b ^{2} - 4ac" ** which is under the square root sign is called the discriminant of the quadratic equation.

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

** Example to calculate discriminant **

**For a quadratic equation x ^{2} + 4x + 4 = 0, determine the determinant**

**Answer**

x^{2} + 5x + 5 = 0

Discriminant = b^{2} – 4ac

Here, a = 1, b = 5, c = 5

**Discriminant = b ^{2} - 4ac = 5^{2} - 4(1)(5) = 5**

Use of discriminant

Discriminant tells about the nature of the roots of the quadratic equation. The roots may be real, equal or imaginary.

Depending upon the value of discriminant, a quadratic equation could have :

◾ **two real and unequal roots** when discriminant **value > 0**

◾ **only one real and equal root** when discriminant **value = 0**

◾ **no real roots ie. both the roots** are imaginary when discriminant **value < 0**

** Case I- When Discriminant > 0**

**For a quadratic equation x ^{2} - 12x + 32 = 0, determine the discriminant and the roots of the quadratic equation**

**Answer**

Discriminant = b^{2} – 4ac

Here, a = 1, b = -12 , c = 32

Discriminant = (-12)^{2} - 4(1)(32) = 144 - 128 = 16

We know that the quadratic formula is

Roots(x_{1}, x_{2}) = |
−b ± √ b2 − 4ac |

2a |

x_{1} = |
−(-12) + √ (-12)2 − 4(1)(32) | = 8 |

2(1) |

x_{1} = |
−(-12) - √ (-12)2 - 4(1)(32) | = 4 |

2(1) |

Hence, for the quadratic equation x^{2} - 12x + 32 = 0, we get two real and distinct roots- 8 and 4

**Hence, for Discriminant > 0, the quadratic equation has 2 real and distinct roots **

** Case II- When Discriminant = 0**

**For a quadratic equation x ^{2} - 12x + 36 = 0, determine the discriminant and the roots of the quadratic equation**

**Answer**

Discriminant = b^{2} – 4ac

Here, a = 1, b = -12 , c = 36

Discriminant = (-12)^{2} - 4(1)(36) = 144 - 144 = 0

We know that the quadratic formula is

Roots(x_{1}, x_{2}) = |
−b ± √ b2 − 4ac |

2a |

x_{1} = |
−(-12) + √ (-12)2 − 4(1)(36) | = 6 |

2(1) |

x_{1} = |
−(-12) - √ (-12)2 - 4(1)(36) | = 6 |

2(1) |

Hence, for the quadratic equation x^{2} - 12x + 36 = 0, we get only one real and equal root, which is 6.

**Hence, for Discriminant = 0, the quadratic equation has only one real and equal root **

** Case III- When Discriminant < 0**

**For a quadratic equation x ^{2} + 4x + 10 = 0, determine the discriminant and the roots of the quadratic equation**

**Answer**

Discriminant = b^{2} – 4ac

Here, a = 1, b = 4 , c = 10

Discriminant = (4)^{2} - 4(1)(10) = 16 - 40 = -24

We know that the quadratic formula is

Roots(x_{1}, x_{2}) = |
−b ± √ b2 − 4ac |

2a |

x_{1} = |
−(4) + √(4)2 − 4(1)(10) | = | −(4) + √ -24 |

2(1) | 2 |

x_{1} = |
−(4) - √(4)2 - 4(1)(10) | = | −(4) - √-24 |

2(1) | 2 |

Hence, for the quadratic equation x^{2} + 4x + 10 = 0, we get no real roots

**Hence, for Discriminant < 0, the quadratic equation has no real roots **

** Summary **

Discriminant value Cases | Roots of quadratic | Factorisation of quadratic |
---|---|---|

Discriminant value > 0 | two real distinct roots | two distinct linear factors |

Discriminant value = 0 | two identical real roots | two identical linear factors |

Discriminant value < 0 | No real roots | Unable to factorise |

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