The **midnight formula**, also known as the abc formula, is a solution formula for mixed-square equations. It owes its name to the fact that it is considered so important that every student should know it by heart even at midnight in their sleep. It is similar to the PQ formula, which can be used to calculate the same results.

The **abc formula** can be used to calculate the zeros of a function, i.e. exactly the points at which y = 0. The function curve intersects the x-axis there. Whether the quadratic equation has one, two or no zeros depends on what is under the root.

## How do you proceed with the calculation?

A quadratic equation is a second-degree equation, i.e. the variable x does not occur in any power higher than the second power.

The general form of a quadratic **equation** is: **ax² bx c = 0**

Please note that a must not be 0.

To solve this equation, the midnight formula is used, which looks like this:

**Positive form**: X1 = -b √b2 - 4ac / 2a**Negative**form: X2 = -b - √b2 - 4ac / 2a

## Discriminant

The discriminant is the term under the root that makes a statement about the solvability of the equation.

**D (discriminant) = b2 - 4ac**

- if D > 0, then there are two different real solutions x1 and x2
- if D = 0, there is one real solution (of multiplicity 2)
- if D < 0, there is no real solution

## Instructions for the calculation

The following is a step-by-step guide to using the midnight formula:

- Step: write down the starting equation
- Step: Set the equation to zero. This means converting it so that there is only a 0 on one side (usually on the right).
- Step: Find out which values for a, b and c can be used in the formula
- Step: Calculating the solution with a in front of the root
- Step: Calculating the solution with a - in front of the root

### Example with two zeros

An example with two zeros is presented below:

- Step: 2x²- 5 = 0 - 3x
- Step: 2x² - 5 = 0 - 3x / 3x = 3x² 3x - 5 = 0
- Step: a = 2, b = 3, c = -5
- Step: X1 = (-3 √32-4x2x (-5)) / 2x2

X1= (-3 √49) /4

X1 = (-3 7) / 4

**X1 = 1**

- Step: X2= (-3 - √32 - 4 * 2x [-5]) / 2x²

X2= (-3 - √49) / 4

X2 = (-3 - 7) / 4

**X2 = -10 / 4 -> 5/2**

### Example without zeros

- Step: 2x² 3x 30 = 0
- Step: 2x² 3x 30 = 0
- Step: a = 2, b = 3, c = 30
- Step: X1 = (-3 √32 - 4x² x 30) / 2x²

X1 = -3 √-231 / 4 --> no solution!

- Step: X2 = (-3 - √32 - 4x² x 30) / 2x²

X2= -3 - √-231 / 4 -> No solution!

## Where is the formula used?

The midnight formula is very important in algebra and is also used for calculations in physics and chemistry in addition to mathematics.

In English, the midnight formula is called the "*quadratic formula"*. A direct translation as "*Midnight formula*" does not exist.

## Mathematics

- Angle
- Ball
- Binomial formula
- Circle
- Cone
- Cube
- Cuboid
- Cylinder
- Derivation rules
- Difference
- Dragon square
- Fractions
- Integral
- Midnight formula
- Parallelogram
- Percent
- Polynomial division
- PQ formula
- Pyramid
- Rectangle
- Rhombus
- Rule of three
- Square
- Standard deviation
- Sum of digits
- Surface area
- Trapezoid
- Triangle
- Volume
- Zeros

## Mathematics

- Angle
- Ball
- Binomial formula
- Circle
- Cone
- Cube
- Cuboid
- Cylinder
- Derivation rules
- Difference
- Dragon square
- Fractions
- Integral
- Midnight formula
- Parallelogram
- Percent
- Polynomial division
- PQ formula
- Pyramid
- Rectangle
- Rhombus
- Rule of three
- Square
- Standard deviation
- Sum of digits
- Surface area
- Trapezoid
- Triangle
- Volume
- Zeros