A quadratic is a type of problem In mathematics that deals with a variable multiplied by itself (an operation known as squaring).

It derives from the area of a square being its side length multiplied by itself. The word quadratic comes from “quadratum”, which means square in Latin word.

There are a lot of phenomena in the real world that can be described by Quadratic equations. For example is rocket fly route, where it will land, how long it will take a person to row down & up a river, or how much to charge for a product.

Because of these applications, quadratics equations have profound historical importance and were foundation of algebra.

Table of Contents

## Quadratic Equation

Only if it can be put in the form *ax*^{2}* + bx + c *= 0, and a is not zero.

The name comes from “quad” meaning square, as the variable is squared (in other words x2).

These are all quadratic equations in disguise:

Equation | In standard form | a, b and c |

x^{2} = x + 1 | x² -x – 3 = 0 | a=1, b=-1, c=-3 |

2(x^{2} – 2x) = 2 | 2x² – 4x – 2 = 0 | a=2, b=-4, c=-2 |

x(x-6) = 18 | -x² +6x + 18 = 0 | a=-1, b=6, c=18 |

20 – 15/x – 10/x^{2} = 0 | 20x² -15x – 10 = 0 | a=20, b=-15, c=-10 |

## Quadratic Formula

How Does this Work?

The solution(s) to a quadratic equation can be calculated using the Quadratic Formula:

The “±” means we need to do a plus AND a minus, so there are normally TWO solutions !

The part **(b2 – 4ac)** is called the “** discriminant**“, because it can “discriminate” between the possible types of answer:

- when b2 – 4ac > 0, we get two real solutions,
- when b2 – 4ac = 0, we get just ONE solution,
- when b2 – 4ac < 0, we get complex solutions.

### Example

Find x !

Equation | a, b and c | Operation | x |

x² -x – 3 = 0 | a=1, b=-1, c=-3 | ||

2x² – 4x – 2 = 0 | a=2, b=-4, c=-2 |

In order to Quadratic Formula to work, you must have your equation arranged in the form:

**(quadratic) = 0**

Also, the 2a in the denominator of the Formula is not just the square root. It is underneath everything above.

Make sure not to drop the square root or the “±” in the middle of your calculations, so you will not need to put them back in.

Remember that b^{2} means “the square of ALL of b, including its sign”. Do not leave b^{2} being negative, even if b is negative, because the square of a negative is a positive.

In other words, don’t be sloppy and don’t try to take shortcuts, because it will only hurt you in the long run. Trust me on this!