A Quadratic Formula Calculator is a tool that can be used to find the roots or solutions of a quadratic equation.

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The Quadratic Formula is:

x = (-b ± sqrt(b^2 - 4ac)) / (2a)

This formula gives the two solutions for the quadratic equation, which are also known as the roots of the equation.

If the discriminant (b^2 - 4ac) is positive, the quadratic equation has two real roots.

If the discriminant is zero, the quadratic equation has one real root (known as a double root).

If the discriminant is negative, the quadratic equation has no real roots (but two complex roots).

The Quadratic Formula can be derived by completing the square of the quadratic equation. The process involves adding and subtracting a term (b^2 / 4a^2) inside the parentheses so that the quadratic expression inside the square root can be simplified into a perfect square. The resulting expression is then solved for "x" using basic algebraic operations.

## What is the quadratic formula?

The quadratic formula calculator is an online tool that is used to calculate the roots of a quadratic equation. The quadratic equation is a second-degree polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable. The quadratic formula is derived from the standard form of a quadratic equation, and it is given by:

x = (-b ± √(b^2 – 4ac)) / 2a

The quadratic formula calculator solves the quadratic equation using this formula and gives the two roots of the equation.

## What is the quadratic equation?

The quadratic equation is a second-degree polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable. It is a fundamental concept in algebra and has numerous applications in science, engineering, economics, and many other fields.

The quadratic equation can have either one or two real roots, or two complex roots, depending on the values of the coefficients a, b, and c. The quadratic formula is used to calculate the roots of the quadratic equation, and it is given by:

x = (-b ± √(b^2 – 4ac)) / 2a

The solutions to the quadratic equation can also be found by factoring, completing the square, or graphing the equation.

## Derivation of the quadratic formula

The quadratic formula is used to solve the quadratic equation ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable. The formula is given by:

x = (-b ± √(b^2 – 4ac)) / 2a

To derive this formula, we start by completing the square of the quadratic equation:

ax^2 + bx + c = 0

ax^2 + bx = -c

We now add and subtract (b/2a)^2 to the left-hand side of the equation:

ax^2 + bx + (b/2a)^2 – (b/2a)^2 = -c

The first three terms on the left-hand side can be factored as a perfect square:

a(x + b/2a)^2 = -c + (b/2a)^2

We can now isolate x:

(x + b/2a)^2 = (-c + (b/2a)^2) / a

Taking the square root of both sides, we obtain:

x + b/2a = ± √((-c + (b/2a)^2) / a)

Subtracting b/2a from both sides gives:

x = (-b ± √(b^2 – 4ac)) / 2a

This is the quadratic formula, which can be used to find the solutions to any quadratic equation of the form ax^2 + bx + c = 0.

## How to use the Quadratic formula calculator?

- Go to the Quadratic Formula Calculator website.
- Enter the values of a, b, and c in the input fields.
- Click on the “Calculate” button to get the solutions.
- The calculator will display the roots of the quadratic equation in the standard form (ax^2 + bx + c = 0).

Note: The calculator gives both the real and complex roots of the equation, if any.

Suppose we have a quadratic equation:

2x^2 + 5x – 3 = 0

We can use the quadratic formula to find the roots of this equation.

First, we identify the values of a, b, and c:

a = 2 b = 5 c = -3

Next, we plug these values into the quadratic formula:

x = (-b ± √(b^2 – 4ac)) / 2a

x = (-5 ± √(5^2 – 4(2)(-3))) / 2(2)

Simplifying the equation, we get:

x = (-5 ± √49) / 4

x = (-5 ± 7) / 4

So the solutions are:

x1 = (-5 + 7) / 4 = 1/2

x2 = (-5 – 7) / 4 = -3/2

Therefore, the roots of the equation 2x^2 + 5x – 3 = 0 are x1 = 1/2 and x2 = -3/2.