Mathematics in Action: Finding Roots of Quadratic Equations
by Yuvi K - December 16, 2023
[ Mathematics in Action: Finding Roots of Quadratic Equations ]
Mathematics is a field of great use and power, which is why it is so important in our lives. Mathematicians and scientists alike use mathematics in countless ways. One such example is in the finding of roots of quadratic equations. In this article, we will take a look at what a quadratic equation is, and how we can find its roots.
What is a Quadratic Equation?
A quadratic equation is a polynomial equation that has the general form:
ax2 + bx + c = 0
Where a, b, and c are real numbers. It is called a quadratic equation because of the term “squared” that is included in the equation, i.e., “ax2”.
Properties of Quadratic Equation
There are few properties that are consistently present in a quadratic equation. These properties are:
- The degree of the equation is 2 (two).
- The equation can have two, one, or zero real roots (दो ,एक , या शून्य).
- The value of the determinant (परिभाषा) must be greater than zero, as it tells the number of roots (रूट) of the equation.
- The equation can be factorised into the product of two linear factors (लीनियर फॅक्टर्स)
- The term with the highest degree is always placed at the left side of the equation.
- The terms are always written in descending order of the powers of x.
- The equation is always in the form of ax2 + bx + c = 0.
General Form of the Quadratic Equation
The general form of the quadratic equation is written as such:
ax2 + bx + c = 0
Where ‘a’, ‘b’, and ‘c’ are real coefficients (रियल कोफीशियन्ट्स). You can also consider it as a standard form.
The Discriminant of the Quadratic Equation
The discriminant of a quadratic equation is the term ‘b2 – 4ac’ in the equation. It helps us to determine the number of roots that the equation has. The discriminant is calculated using the ‘delta’ symbol, which is written as such Δ = b2 – 4ac. The value of the discriminant determines the number of roots that the quadratic equation has as follows:
| Δ | Number of Roots (रूट) |
|---|---|
| Δ > 0 | 2 Real Roots (दो रियल रूट) |
| Δ = 0 | 1 Real Root (एक रियल रूट) |
| Δ < 0 | No Real Roots (शून्य रियल रूट) |
Finding the Roots of a Quadratic Equation
In order to find the roots of a quadratic equation, we must first rewrite it in the form ax2 + bx + c = 0 which is known as the standard form. Then, we must use the Quadratic Formula, which is an equation that can be used for solving the quadratic equation. The Quadratic Formula is written as follows:
x1, x2 = -b ± √Δ / 2a
Where ‘b’ and ‘a’ are constants that remain unchanged, and ‘ Δ’ is the discriminant.
The Quadratic Formula can be simplified as such:
x1, x2 = (-b ± √Δ) / 2a
If we have a = 1, the equation can be further simplified to:
x1 = -b + √Δ
x2 = -b – √Δ
Example: Finding the Roots of a Quadratic Equation
Let us consider the equation:
3x2 + 5x + 2 = 0
We first rewrite the equation in the standard form:
3x2 + 5x + 2 = 0
Then, we calculate the discriminant:
Δ = b2 – 4ac
=> Δ = 52 – 4(3)(2)
=> Δ = 25
Using the Quadratic Formula, we can calculate the roots of the equation as follows:
x1, x2 = (-b ± √Δ) / 2a
=> x1, x2 = (-5 ± √25) / 6
=> x1 = -5 + √25 / 6
=> x1 = 1/3
=> x2 = -5 – √25 / 6
=> x2 = -7/3
Hence, the roots of the equation 3x2 + 5x + 2 = 0 are x1 = 1/3 and x2 = -7/3.
Conclusion
In this article, we have discussed what a quadratic equation is, its properties, and how to find its roots. The Quadratic Formula is the most widely used method for solving quadratic equations, and is a powerful tool for mathematicians. We hope that this article has helped you gain a better understanding of this fascinating mathematics topic.
