GoMim AI | What is quadratic formula and How to Calculate it

Introduction

The quadratic formula is a fundamental tool in algebra that allows us to find solutions to quadratic equations. These equations appear frequently in various fields, including physics, engineering, and finance. Understanding how to use the quadratic formula can unlock a deeper comprehension of mathematical concepts and problem-solving techniques.

What is it?

The quadratic formula is used to solve quadratic equations, which are equations of the form $$ax^2 + bx + c = 0$$ Here, 'a', 'b', and 'c' are constants, and 'x' represents the variable we need to solve for. The quadratic formula is expressed as: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ This formula provides the solution(s) for 'x' by calculating the values that satisfy the equation.

Why is it important?

The quadratic formula is crucial for solving quadratic equations, which are prevalent in many areas of study and practical applications. In academics, these equations often appear in math exams and are essential for students to master. In real-world applications, quadratic equations model various phenomena such as projectile motion in physics, the shape of parabolic mirrors in engineering, and even in calculating interest rates in finance. Understanding and being able to apply the quadratic formula is therefore a valuable skill that extends beyond the classroom.

How to Calculate it Step-by-Step

To calculate the roots of a quadratic equation using the quadratic formula, follow these steps:

1. Identify the coefficients: Determine the values of 'a', 'b', and 'c' in the quadratic equation.

2. Calculate the discriminant: Use the formula $$b^2 - 4ac$$ to find the discriminant. This value will help determine the nature of the roots.

3. Apply the quadratic formula: Substitute 'a', 'b', and 'c' into the quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

4. Solve for 'x': Compute the values of 'x' by evaluating the expression using both the '+' and '−' operations.

Example: Solve the quadratic equation $$2x^2 + 3x - 2 = 0$$

- Step 1: Identify coefficients: a = 2, b = 3, c = -2.

- Step 2: Calculate the discriminant: $$3^2 - 4(2)(-2) = 9 + 16 = 25$$

- Step 3: Apply the formula: $$x = \frac{-3 \pm \sqrt{25}}{4}$$

- Step 4: Solve for 'x':

- $$x = \frac{-3 + 5}{4} = \frac{2}{4} = 0.5$$

- $$x = \frac{-3 - 5}{4} = \frac{-8}{4} = -2$$ Thus, the solutions are x = 0.5 and x = -2.

Related Practice Problem

Problem: Solve the quadratic equation $$3x^2 - 6x + 2 = 0$$ using the quadratic formula.

Step-by-step Solution:

1. Identify the coefficients: a = 3, b = -6, c = 2.

2. Calculate the discriminant: $$(-6)^2 - 4(3)(2) = 36 - 24 = 12$$

3. Apply the quadratic formula: $$x = \frac{-(-6) \pm \sqrt{12}}{2(3)}$$

4. Simplify under the square root: $$\sqrt{12} = 2\sqrt{3}$$

5. Solve for 'x':

- $$x = \frac{6 + 2\sqrt{3}}{6} = 1 + \frac{\sqrt{3}}{3}$$

- $$x = \frac{6 - 2\sqrt{3}}{6} = 1 - \frac{\sqrt{3}}{3}$$

Therefore, the solutions are $$x = 1 + \frac{\sqrt{3}}{3}$$ and $$x = 1 - \frac{\sqrt{3}}{3}$$

Use GoMim Math AI Solver for quadratic formula

To quickly and accurately solve quadratic equations like those discussed above, you can use the GoMim Math AI Solver at gomim.com. This AI tool simplifies the process by providing instant solutions, saving time and effort. Try it now!

FAQ

Q: What is the discriminant in a quadratic equation?

A: The discriminant is the part of the quadratic formula under the square root, expressed as $$b^2 - 4ac$$ It determines the nature of the roots of the equation.

Q: Can the quadratic formula be used for all quadratic equations?

A: Yes, the quadratic formula can solve any quadratic equation, regardless of whether the roots are real or complex.

Q: What does it mean if the discriminant is negative?

A: A negative discriminant indicates that the quadratic equation has two complex (or imaginary) roots.

Q: How does the quadratic formula calculator work?

A: A quadratic formula calculator automates the calculation by taking inputs for 'a', 'b', and 'c', and computing the roots using the quadratic formula.

Q: What are the benefits of using a math AI solver for quadratic equations?

A: A math AI solver provides quick, accurate solutions, helps verify manual calculations, and allows for efficient learning and problem-solving.

Conclusion

Understanding the quadratic formula is essential for solving quadratic equations, a common component of mathematics education and various real-world applications. By mastering this formula, students and professionals can enhance their problem-solving capabilities. Additionally, tools like the GoMim Math AI Solver can greatly assist in learning and applying the quadratic formula efficiently.