Roots of a Quadratic Equation Calculator

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Lowest Common Multiple (LCM)

$\bold{Table \space of \space Content}$

• 1. Introduction to the Quadratic Equation & its roots.
• 2. What is the Formulae used ?
• 3. How do I calculate the roots of a quadratic equation ?
• 4. Why choose our roots of a quadratic equation Calculator?
• 5. A Video for explaining this concept
• 6. How to use this calculator ?
• 7. Solved Examples
• 8. Frequently Asked Questions (FAQs)
• 9. What are the real-life applications?
• 10. Conclusion

1. Introduction to the Quadratic Equation & its roots

$\bold{Quadratic \space Equation}$
In mathematics, a quadratic equation is defined as a second-degree polynomial equation of the form: $\bold{ax^2 + bx + c = 0}$, where a $\ne 0$, 'a', 'b', and 'c' are constants, and 'x' is a variable. The roots of a quadratic equation are the values of 'x' that satisfy the equation, making it equal to zero.
$\bold{Roots \space of \space a \space Quadratic \space equation}$
The roots of a quadratic equation are the values of 'x' that satisfy the equation, making it equal to zero. Roots can be real or complex numbers depending on the values of a, b, and c.

2. What is the Formulae used?

We can find the roots of a quadratic equation by using the DISCRIMINANT Method or Shridhar Acharya ji Method (An Indian Scientist).
$\bold{x = \frac{-b \space \pm \space \sqrt{b^2 - 4ac}}{2a}}$ where a is the coefficient of $x^2$, b is the coefficient of x and c is the constant term.
We can check the NATURE of the roots by finding the value of Discrimiat i.e. $\bold{D = b^2 -4ac}$
if $\bold{D > 0}$ then the Roots of the quadratic equation are $\bold{REAL \space and \space DISTINCT}$.
if $\bold{D = 0}$ then the Roots of the quadratic equation are $\bold{REAL \space and \space EQUAL}$.
if $\bold{D < 0}$ then the Roots of the quadratic equation are $\bold{IMAGINARY \space and \space COMPLEX \space CONJUGATE}$.

3. How do I calculate the distance between two points?

Write the values a, b, and c from the quadratic equation.
Calculate the value of "D" to check the Nature of the roots.
Put the values in the above-given formula and calculate the value of the roots.

4. Why choose our Addition/Subtraction of two Vectors Calculator?

$\bold{Easy \space \space to \space Use}$
Our calculator page provides a user-friendly interface that makes it accessible to both students and professionals. You can quickly input your square matrix and obtain the matrix of minors within a fraction of a second.

$\bold{Time \space Saving \space By \space automation}$
Our calculator saves you valuable time and effort. You no longer need to manually calculate each cofactor, making complex matrix operations more efficient.

$\bold{Accuracy \space and \space Precision}$
Our calculator ensures accurate results by performing calculations based on established mathematical formulas and algorithms. It eliminates the possibility of human error associated with manual calculations.

$\bold{Versatility}$
Our calculator can handle all input values like integers, fractions, or any real number.

$\bold{Complementary \space Resources}$
Alongside this calculator, our website offers additional calculators related to Pre-algebra, Algebra, Precalculus, Calculus, Coordinate geometry, Linear algebra, Chemistry, Physics, and various algebraic operations. These calculators can further enhance your understanding and proficiency.

5. A video based on finding the roots of a quadratic equation.

6. How to use this calculator

This calculator will help you to find the value of the roots of the quadratic equation.
In the given input boxes, you have to put the value for a, b, and c.
After clicking on the Calculate button, a step-by-step solution will be displayed on the screen.
You can access, download, and share the solution.

7. Solved Examples

$\bold{Question:1}$
By using the discriminant method, Find the roots of the equation $x^2$ + 3x + 2.
$\bold{Solution}$
By comparing the given equation with $ax^2$ + bx + c = 0 we can determine the values of a, b and c.
a = 1, b = 3, c = 2
Since value of D = $b^2$ - 4ac = $3^2$ - 4.1.2 = 1 > 0 then Roots are real & distinct.
Roots = $\frac{-b \pm \sqrt{b^2 - 4ac} }{2a}$ = $\frac{-3 \pm \sqrt{3^2 - 4.1.2} }{2.1}$
after solving the above equation roots are $\alpha$ = -2 & $\beta$ = -1

$\bold{Question:2}$
By using the discriminant method, Find the roots of the equation $2x^2$ + x + 1.
$\bold{Solution}$
By comparing the given equation with $ax^2$ + bx + c = 0 we can determine the values of a, b, and c.
a = 2, b = 1, c = 1
Since value of D = $b^2$ - 4ac = $1^2$ - 4.2.1 = -7 < 0 then Roots are imaginary & complex conjugate.
Roots = $\frac{-b \pm \sqrt{b^2 - 4ac} }{2a}$ = $\frac{-1 \pm \sqrt{1^2 - 4.2.1} }{2.2}$
after solving the above equation roots are $\alpha$ = $\frac{-1 + \sqrt{7}i}{4}$ & $\beta$ = $\frac{-1 - \sqrt{7}i}{4}$

8. Frequently Asked Questions (FAQs)

What is a quadratic equation calculator, and how does it work?

A quadratic equation calculator is a tool used to find a quadratic equation's roots (solutions). It works by taking the equation coefficients (a, b, and c) as input and applying the quadratic formula to calculate the roots.

What is the quadratic formula, and why is it important?

The quadratic formula is a fundamental mathematical formula used to find the roots of a quadratic equation. It is important because it provides a reliable and systematic way to solve quadratic equations with various real-life applications.

Can a quadratic equation have complex roots, and can a calculator handle them?

Yes, quadratic equations can have complex roots, especially when the discriminant (b^2 - 4ac) is negative. Most quadratic equation calculators can handle complex roots and will provide them in the form of complex numbers (a + bi).

Are there different methods to solve quadratic equations besides calculators?

Yes, alternative methods exist to solve quadratic equations, such as factoring, completing the square, and graphing. These methods can be useful when you prefer not to use a calculator or want to understand the concept better.

What are some common mistakes to avoid when using a quadratic equation calculator?

Common mistakes include entering the coefficients incorrectly, forgetting to use the negative sign before the b coefficient when inputting it, and not considering the discriminant to determine the nature of the roots (real, complex, or equal).

Can I use a quadratic equation calculator for real-world problem-solving?

Absolutely! Quadratic equations are used in various real-life applications, such as physics (for motion and trajectories), engineering (for optimization problems), and finance (for modeling investments and loans). A calculator makes solving these problems faster and more accurate.

Is it necessary to understand the underlying math to use a quadratic equation calculator effectively?

While you can use a quadratic equation calculator without a deep understanding of the math involved, having a basic grasp of the quadratic formula and its components can help you interpret the results and use them effectively in real-life scenarios.

What are the real-life applications?

It has vast applications in Projectile Motion, Engineering and Optimization, Finance, Geometry, Computer Graphics, Environmental Science, Architecture, etc.

Conclusion

In conclusion, a root of a quadratic equation calculator is a valuable tool for solving a wide range of real-world problems across various fields. It simplifies the process of finding the origins of quadratic equations, which are prevalent in mathematics and numerous applications. Whether you're an engineer, physicist, finance professional, or simply a student learning algebra, understanding how to use this calculator effectively can save time and enhance problem-solving abilities in your respective domain.

This blog is written by Neetesh Kumar

If you have any suggestions regarding the improvement of the content of this page, please write to me at My Official Email Address: [email protected]

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