Calculator

Quadratic Formula Calculator – Free Online Solver

Quadratic Formula Calculator – Free Online Solver

Q: 4. What are the different methods for solving quadratic equations besides the quadratic formula?

Other methods include factoring, completing the square, and graphical methods. Factoring involves expressing the quadratic expression as a product of two linear factors. Completing the square transforms the equation into a perfect square trinomial, making it easier to solve. Graphical methods involve plotting the quadratic equation and finding the x-intercepts (roots). The quadratic formula, however, provides a universal solution for all quadratic equations.

Q: 5. Can the quadratic formula be used to solve equations that are not in standard form (ax² + bx + c = 0)?

Yes, but the equation must first be rearranged into standard form (ax² + bx + c = 0) before applying the quadratic formula. This involves moving all terms to one side of the equation, ensuring that the equation is equal to zero.

Q: 6. What happens if the discriminant is negative in the quadratic formula?

A negative discriminant indicates that the quadratic equation has two complex roots, which are not real numbers. Complex roots are expressed in the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit (√-1).

Q: 7. How can I check my answers after solving a quadratic equation using the quadratic formula?

Substitute the calculated roots (values of x) back into the original quadratic equation. If the equation holds true (both sides are equal), then the solutions are correct.

Q: 8. What are some real-world applications of solving quadratic equations?

Quadratic equations are used extensively in various fields, including: Physics: Calculating projectile motion, determining the trajectory of an objectEngineering: Designing curves, bridges, and other structuresBusiness: Finding the maximum profit or minimum costEconomics: Modeling supply and demand

Q: 9. What if 'a' is equal to zero in a quadratic equation?

If 'a' is zero, the equation is no longer quadratic; it becomes a linear equation. The quadratic formula cannot be applied in this case. The equation can then be solved using simpler algebraic methods.

Q: 10. Are there any limitations to using the quadratic formula?

The main limitation is that it only applies to quadratic equations. It cannot be used to solve higher-degree polynomial equations or other types of equations. Also, if the coefficients a, b, and c are very large or very small numbers, rounding errors could affect the accuracy of the calculated roots.

How to Find the Roots of a Quadratic Equation Using the Calculator

Quadratic Formula Calculator

What is Quadratic Formula Calculator?

The Quadratic Formula Calculator is an online tool that finds solutions of any quadratic equation of the form ax² + bx + c = 0. It allows you to enter the coefficients a, b, and c, then instantly calculates the roots of the equation, showing both real and complex solutions with detailed solving steps.

Formula or Logic Behind Quadratic Formula Calculator

The calculator uses the standard quadratic formula:
x = [ -b ± √(b² − 4ac) ] / (2a) Here, ‘b² − 4ac’ is called the discriminant, denoted as D. Depending on the value of D, the equation has two real roots, one double (real) root, or two complex roots. The calculator checks these cases and computes answers accordingly.

Quadratic Equation Examples and Solutions

Equation (ax² + bx + c = 0)	a	b	c	Roots (x)
2x² + 4x - 6 = 0	2	4	-6	1 and -3
x² - 6x + 9 = 0	1	-6	9	3 (double root)
x² + 2x + 5 = 0	1	2	5	-1 + 2i, -1 - 2i
x² + 8x + 16 = 0	1	8	16	-4 (double root)
3x² - x - 4 = 0	3	-1	-4	1.3333, -1

Steps to Use the Quadratic Formula Calculator

Enter the required number or values for coefficients a, b, and c in the input boxes
Click on the 'Calculate' button
Get instant results with solution steps and root types (real or complex)

Why Use Vedantu’s Quadratic Formula Calculator?

Easy to use, mobile-friendly, and designed for students and professionals needing quick, reliable solutions. Vedantu's calculator displays step-by-step workings, making it perfect for learning and exam preparation. Trusted by millions, it supports accurate answers for all types of quadratic equations.

Real-life Applications of Quadratic Formula Calculator

Quadratic equations appear in projectile motion, area optimization problems, physics, engineering, business profit/loss modeling, and daily problem-solving—like calculating break-even points or predicting paths of moving objects. The calculator saves time, helps verify answers, and aids in learning through instant feedback.

For deeper algebra practice, also check the Completing the Square Calculator and Discriminant Calculator to explore alternate methods.
Need to solve other equations? Try the Fraction Calculator and Quadratic Equations Explained for detailed concepts and worked questions on Vedantu.

FAQs on Quadratic Formula Calculator – Free Online Solver

1. What is the quadratic formula and how is it used to solve quadratic equations?

The quadratic formula is a method for solving quadratic equations of the form ax² + bx + c = 0, where a, b, and c are constants. It provides a direct way to find the values of x that satisfy the equation. The formula itself is: x = [-b ± √(b² - 4ac)] / 2a. Using this formula, we substitute the values of a, b, and c from the given quadratic equation to calculate the roots (solutions) for x.

2. How do I use the quadratic formula calculator to solve a quadratic equation?

Simply input the values of coefficients a, b, and c into the respective fields of the quadratic formula calculator. Click on the 'Calculate' button, and the calculator will instantly compute the roots of the equation, along with a step-by-step solution demonstrating the application of the quadratic formula.

3. What is the discriminant in the quadratic formula, and what does it tell us?

The discriminant (D) in the quadratic formula is the expression b² - 4ac. It determines the nature of the roots (solutions) of the quadratic equation. If D > 0, there are two distinct real roots; if D = 0, there is one real root (a repeated root); and if D < 0, there are two complex roots.

4. What are the different methods for solving quadratic equations besides the quadratic formula?

Other methods include factoring, completing the square, and graphical methods. Factoring involves expressing the quadratic expression as a product of two linear factors. Completing the square transforms the equation into a perfect square trinomial, making it easier to solve. Graphical methods involve plotting the quadratic equation and finding the x-intercepts (roots). The quadratic formula, however, provides a universal solution for all quadratic equations.

5. Can the quadratic formula be used to solve equations that are not in standard form (ax² + bx + c = 0)?

Yes, but the equation must first be rearranged into standard form (ax² + bx + c = 0) before applying the quadratic formula. This involves moving all terms to one side of the equation, ensuring that the equation is equal to zero.

6. What happens if the discriminant is negative in the quadratic formula?

A negative discriminant indicates that the quadratic equation has two complex roots, which are not real numbers. Complex roots are expressed in the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit (√-1).

7. How can I check my answers after solving a quadratic equation using the quadratic formula?

Substitute the calculated roots (values of x) back into the original quadratic equation. If the equation holds true (both sides are equal), then the solutions are correct.

8. What are some real-world applications of solving quadratic equations?

Quadratic equations are used extensively in various fields, including:

Physics: Calculating projectile motion, determining the trajectory of an object
Engineering: Designing curves, bridges, and other structures
Business: Finding the maximum profit or minimum cost
Economics: Modeling supply and demand

9. What if 'a' is equal to zero in a quadratic equation?

If 'a' is zero, the equation is no longer quadratic; it becomes a linear equation. The quadratic formula cannot be applied in this case. The equation can then be solved using simpler algebraic methods.

10. Are there any limitations to using the quadratic formula?

The main limitation is that it only applies to quadratic equations. It cannot be used to solve higher-degree polynomial equations or other types of equations. Also, if the coefficients a, b, and c are very large or very small numbers, rounding errors could affect the accuracy of the calculated roots.

11. How do I interpret the roots obtained from the quadratic formula?

The roots represent the x-intercepts of the parabola represented by the quadratic equation. They are the values of x for which the quadratic expression is equal to zero. If the roots are real, they indicate where the parabola intersects the x-axis. If they are complex, the parabola does not intersect the x-axis.

12. What is the relationship between the roots of a quadratic equation and its coefficients?

There's a relationship between the sum and product of the roots and the coefficients of the quadratic equation. If α and β are the roots, then: Sum of roots (α + β) = -b/a; Product of roots (αβ) = c/a. These relationships can be used to verify the solutions obtained using the quadratic formula.