# Quadratic Formula Calculator

Effortlessly solve quadratic equations using our Quadratic Formula Calculator. Find the roots and solutions with ease, simplifying complex calculations. Accelerate your problem-solving in mathematics, physics, and engineering.
The calculator below solves the quadratic equation of ax^{2} + bx + c = 0.

a = |
b = |
c = |
x = |

Fractional values such as 3/4 can be used. |

In algebra, a quadratic equation is any polynomial equation of the second degree with the following form:

ax^{2} + bx + c = 0

where **x** is an unknown, **a** is referred to as the quadratic coefficient, **b** the linear coefficient, and **c** the constant. The numerals **a**, **b**, and **c** are coefficients of the equation, and they represent known numbers. For example, **a** cannot be 0, or the equation would be linear rather than quadratic. A quadratic equation can be solved in multiple ways, including factoring, using the quadratic formula, completing the square, or graphing. Only the use of the quadratic formula, as well as the basics of completing the square, will be discussed here (since the derivation of the formula involves completing the square). Below is the quadratic formula, as well as its derivation.

### Derivation of the Quadratic Formula

From this point, it is possible to complete the square using the relationship that:

x^{2} + bx + c = (x - h)^{2} + k

Continuing the derivation using this relationship:

Recall that the ± exists as a function of computing a square root, making both positive and negative roots solutions of the quadratic equation. The **x** values found through the quadratic formula are roots of the quadratic equation that represent the **x** values where any parabola crosses the x-axis. Furthermore, the quadratic formula also provides the axis of symmetry of the parabola. This is demonstrated by the graph provided below. Note that the quadratic formula actually has many real-world applications, such as calculating areas, projectile trajectories, and speed, among others.

### Quadratic Formula Calculator Example

Quadratic Equation | Coefficient 'a' | Coefficient 'b' | Coefficient 'c' | Solutions |
---|---|---|---|---|

2x^2 - 5x + 2 | 2 | -5 | 2 | ? |

x^2 + 6x + 9 | 1 | 6 | 9 | ? |

4x^2 + 4x + 1 | 4 | 4 | 1 | ? |

In this example, we have three different quadratic equations that we want to solve using the Quadratic Formula Calculator.

To solve a quadratic equation using the quadratic formula:

- Identify the coefficients 'a', 'b', and 'c' from the given equation.
- Substitute the values of 'a', 'b', and 'c' into the quadratic formula: x = (-b Β± β(b^2 - 4ac)) / (2a).
- Perform the necessary calculations to find the solutions for 'x'.

Let's solve each quadratic equation:

- For the quadratic equation 2x^2 - 5x + 2:

- Coefficient 'a' = 2, coefficient 'b' = -5, coefficient 'c' = 2.
- Substituting these values into the quadratic formula, we have:

`ε€εΆδ»£η ````
x = (-(-5) Β± β((-5)^2 - 4 * 2 * 2)) / (2 * 2)
x = (5 Β± β(25 - 16)) / 4
x = (5 Β± β9) / 4
```

- Simplifying further, we get two solutions:
- x = (5 + 3) / 4 = 8 / 4 = 2.
- x = (5 - 3) / 4 = 2 / 4 = 0.5.

- For the quadratic equation x^2 + 6x + 9:

- Coefficient 'a' = 1, coefficient 'b' = 6, coefficient 'c' = 9.
- Substituting these values into the quadratic formula, we have:

`ε€εΆδ»£η ````
x = (-6 Β± β(6^2 - 4 * 1 * 9)) / (2 * 1)
x = (-6 Β± β(36 - 36)) / 2
x = (-6 Β± β0) / 2
```

- Simplifying further, we get one solution:
- x = -6 / 2 = -3.

- For the quadratic equation 4x^2 + 4x + 1:

- Coefficient 'a' = 4, coefficient 'b' = 4, coefficient 'c' = 1.
- Substituting these values into the quadratic formula, we have:

`ε€εΆδ»£η ````
x = (-4 Β± β(4^2 - 4 * 4 * 1)) / (2 * 4)
x = (-4 Β± β(16 - 16)) / 8
x = (-4 Β± β0) / 8
```

- Simplifying further, we get one solution:
- x = -4 / 8 = -0.5.

Using this method, we can solve quadratic equations and find their corresponding solutions. The table will now look as follows:

Quadratic Equation | Coefficient 'a' | Coefficient 'b' | Coefficient 'c' | Solutions |
---|---|---|---|---|

2x^2 - 5x + 2 | 2 | -5 | 2 | 0.5, 2 |

x^2 + 6x + 9 | 1 | 6 | 9 | -3 |

4x^2 + 4x + 1 | 4 | 4 | 1 | -0.5 |

Thus, the solutions for the quadratic equations are as given in the table above.