# The Concept of Quadratic Equations

*I am a PhD student of mathematics. I have complete MS in math from the University of Pakistan and have been writing online since 2020.*

## Concept of Quadratic Equations in Mathematics.

**Introduction:**

Algebraic expressions of order "2" are called quadratic equations. They belong to the category of higher order equations. The Latin word quadratus, which means "square," is the origin of the word quadratic in the term "quadratic equations." As a result, we refer to equations with a second-degree variable as quadratic equations. As a result, they are often referred to as "Equations of degree " 2."

Quadratic equations have been used in real-world situations from at least 2050 BC, according to historical records. Mathematicians in Babylonia, Egypt, Greece, China, and India are known to have employed geometric techniques to solve quadratic problems. The methods used by different historians to discover new formulas for solving quadratic equations have changed over time.

The first book in Europe with a complete solution to the general quadratic equation was written by a Jewish mathematician named Abraham bar Hiyya Ha-Nasi in Spain around the 12th century. Later, in 1637, René Descartes wrote La Géométrie, it included the complete solution for the quadratic equation that we still use today.

**Formula or Method for Solving the Quadratic Equations:**

Students can easily handle complex numerical problems with the use of formulas for solving quadratic equations. The general formula for quadratic equations is provided here. It is described as follows:

Let the quadratic equation ax^{2}+ bx + c = 0. Where a, b and c are unknown constants and x is unknown variable its value is determined by this formula:

### Formula for Find the Value of unmown variable x:

x = (-b ± √b^{2}- 4ac) / 2a

By put the value of unknown constants we can find the value x. The factor

b^{2}- 4ac is called the discriminant of quadratic equations. Here is example to help you better understand the steps involved in solving quadratic equations:

**Example:**

**To Solve the Quadratic Equation:**

x^{2}+ 3x + 2 = 0

**Solution:**

Given that:

a=1, b=3 and c=2, and

Discriminant = b^{2} − 4ac = (3)^{2} − 4×1×2 = 9 – 8 = 1

Using the quadratic equation formula, x = (−3 ± √1)/2 = −3 / 2

Therefore, x = −3 / 2

** Roots of Quadratic Equation:**

Many students worry whether there can be more than one solution to a quadratic equation. Is it possible for an equation to have no real solutions? The concept of the quadratic equation's root makes it possible; the solution or root of a quadratic equation is the value of a variable for which the equation is satisfied. A polynomial equation's degree is equal to the number of roots.

There are so two roots to a quadratic equation. Let α and β be the two roots of the quadratic equation ax^{2} + bx + c = 0. You may express the formulae for solving quadratic equations as:

(-b-√b^{2}-4ac) / 2a. Here, a, b, and c are real and rational. As a result, the value or expression (b2 - 4ac) under the square root sign determines the nature of the roots and of the equation ax2 + bx + c = 0. We say this because there is no real number that can be the root of a negative number. Consider the quadratic equation x^{2}= -1. There isn't a real number with a negative square. There are therefore no real number solutions to this equation.

**Type of Roots of Quadratic Equations:**

The general formulas for solving quadratic equations are:

α = (-b-b^{2}-4ac)/2a and β = (-b+b^{2}-4ac)/2a

### Type No. 1: b^{2}– 4ac > 0

The roots and of the quadratic equation ax^{2}+ bx+ c = 0 are real and equal, when a, b, and c are real numbers and a ≠ 0 and the discriminant is positive.

### Type No. 2: b^{2}– 4ac = 0

The quadratic equation ax^{2}+ bx+ c = 0 has real and equal roots when a, b, and c are real numbers, a ≠ 0 and the discriminant is zero.

### Type No. 3: b2– 4ac < 0

The quadratic equation ax^{2}+ bx+ c = 0 have unequal and imaginary roots when a, b, and c are real numbers, a ≠ 0, and the discriminant is negative. We refer to the roots in the form of complex numbers.

### Type No. 4: b2– 4ac = 0

ax^{2}+ bx+ c = 0 has real, rational, and unequal roots when a, b, and c are real numbers, a ≠ 0, and the discriminant is positive and perfect square.

### Type No.5: b2 – 4ac > 0 and not perfect square:

The roots of quadratic equation ax^{2}+ bx+ c = 0 are real, irrational, and unequal when a, b, and c are real numbers, a ≠ 0, and the discriminant is positive but not a perfect square. Here the roots α and β form a pair of irrational conjugates.

### Type No.5: b2 – 4ac > 0 is perfect square and a or b is irrational:

The roots of the quadratic equation ax2 + bx + c = 0 are irrational if all three variables a, b, and c are real numbers, a ≠ 0, and the discriminant is a perfect square.

## Summary of Nature of Roots of Quadratic Equations:

Quadratic Equation | Nature of Roots |
---|---|

b2– 4ac > 0 | Real and unequal |

b2 – 4ac = 0 | Real and equal |

b2 – 4ac > 0 (is a perfect square) | Real, rational and unequal |

b2 – 4ac < 0 | Unequal and Imaginary |

b2 – 4ac > 0 (is a perfect square) | Real, irrational and unequal. |

b2 – 4ac > 0 (is a perfect square and a or b is irrational) | Irrational |

*This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.*

**© 2022 Kinza Javaid**