Quadratic formula

There are certain cases when solving a quadratic equation by factorisation or completing the square is time consuming, lengthy or difficult.

In such cases, we use the quadratic formula to solve the quadratic equation.

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, the quadratic formula is:

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

2a |

Step by step process to solve the quadratic equation by quadratic formula:

**Step I: ** Arrange the Quadratic Equation in the standard form of ax^{2} + bx + c = 0.

**Step II: ** Compare the quadratic equation which is to be solved with the standard quadratic equation and find out the values of the coefficients a, b, c.

**Step III: ** Put these values of a, b, c in Quadratic formula.

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

2a |

As you can notice in the formula. This implies, Quadratic formula calculates two values of x: x_{1} and x_{2}, where

x_{1} = |
−b + √ b2 − 4ac |

2a |

x_{2} = |
−b - √ b2 − 4ac |

2a |

**These two values of x for which ax ^{2} + bx + c = 0 holds true, are called solutions of Quadratic Equation,or roots of Quadratic Equation.**

**Example:** Solve, x^{2} - 10x = -24

**Step I:** - Convert the Quadratic Equation you wish to solve in Standard Form of Quadratic Equation, ax^{2} + bx + c = 0

x^{2} -10x = -24 is converted into x^{2} - 10x + 24 = 0

**Step II:** - Find value of coefficient a, b and c by comparing it with Standard Form of Quadratic Equation ax^{2} + bx + c = 0

Comparing x^{2} - 10x + 24 = 0 with ax^{2} + bx + c = 0, we get

a = 1,

b = -10,

c = 24

**Step III:** Putting values of a, b and c in the formula

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

2a |

x_{1} = |
−b + √ b2 − 4ac |

2a |

= | −(-10) + √ (-10)2 − 4(1)(24) | = 6 |

2(1) |

x_{2} = |
−b - √ b2 − 4ac |

2a |

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

2(1) |

6 and 4 are the roots of the quadratic equation x^{2} -10x = -24

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