Quadratic Formula From Roots Calculator

The Quadratic Formula from Roots Calculator helps you determine the quadratic equation when you know its roots. This is useful when you need to reconstruct a quadratic equation from its solutions or when studying polynomial functions.

What is the Quadratic Formula?

The quadratic formula is a standard method for solving quadratic equations of the form:

ax² + bx + c = 0

The standard quadratic formula to find the roots (solutions) of this equation is:

x = [-b ± √(b² - 4ac)] / (2a)

This formula is derived from completing the square, a method of solving quadratic equations by manipulating the equation into a perfect square trinomial.

How to Find the Roots of a Quadratic Equation

To find the roots of a quadratic equation, you can use the quadratic formula. Here are the steps:

Identify the coefficients a, b, and c in the equation ax² + bx + c = 0.
Calculate the discriminant (D) using D = b² - 4ac.
If D is positive, there are two real roots. If D is zero, there is one real root. If D is negative, there are no real roots.
Apply the quadratic formula to find the roots.

For example, if you have the equation x² - 5x + 6 = 0, the roots are x = 2 and x = 3.

Deriving the Quadratic Formula from Roots

If you know the roots of a quadratic equation, you can derive the quadratic equation using the following steps:

Let the roots be r₁ and r₂.
The quadratic equation can be written in its factored form as (x - r₁)(x - r₂) = 0.
Expand the factored form to get the standard quadratic equation: x² - (r₁ + r₂)x + r₁r₂ = 0.

This shows that the sum of the roots (r₁ + r₂) is equal to -b/a, and the product of the roots (r₁r₂) is equal to c/a.

Note: The quadratic formula can also be derived using completing the square, which involves manipulating the equation to form a perfect square trinomial.

Example Calculation

Let's say you have a quadratic equation with roots at x = 2 and x = 3. To find the quadratic equation, follow these steps:

Write the factored form: (x - 2)(x - 3) = 0.
Expand the equation: x² - 5x + 6 = 0.

So, the quadratic equation is x² - 5x + 6 = 0.

You can verify this by plugging the roots back into the equation:

For x = 2: (2)² - 5(2) + 6 = 4 - 10 + 6 = 0.
For x = 3: (3)² - 5(3) + 6 = 9 - 15 + 6 = 0.

Frequently Asked Questions

What is the quadratic formula used for?: The quadratic formula is used to find the roots of a quadratic equation. It provides the solutions to equations of the form ax² + bx + c = 0.
How do you derive the quadratic formula from roots?: You can derive the quadratic formula from roots by starting with the factored form (x - r₁)(x - r₂) = 0 and expanding it to get the standard quadratic equation.
What is the relationship between the roots and the coefficients of a quadratic equation?: The sum of the roots (r₁ + r₂) is equal to -b/a, and the product of the roots (r₁r₂) is equal to c/a in the quadratic equation ax² + bx + c = 0.
Can the quadratic formula be used for non-real roots?: Yes, the quadratic formula can be used for non-real roots. If the discriminant (b² - 4ac) is negative, the roots will be complex conjugates.
What is the discriminant in the quadratic formula?: The discriminant is the part of the quadratic formula under the square root (b² - 4ac). It determines the nature of the roots: positive for two real roots, zero for one real root, and negative for no real roots.