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Root Calculator Quadratic Formula

Reviewed by Calculator Editorial Team

The Quadratic Formula is a fundamental tool in algebra for solving quadratic equations. This calculator helps you find the roots of any quadratic equation in the form ax² + bx + c = 0.

What is the Quadratic Formula?

The Quadratic Formula is a standard method for finding the roots of a quadratic equation. A quadratic equation is any equation that can be written in the form:

ax² + bx + c = 0

Where a, b, and c are constants, and a ≠ 0. The roots of the equation are the values of x that satisfy the equation. The Quadratic Formula provides a direct method to calculate these roots:

x = [-b ± √(b² - 4ac)] / (2a)

The term under the square root (b² - 4ac) is called the discriminant. The discriminant tells us about the nature of the roots:

  • If the discriminant is positive, there are two distinct real roots.
  • If the discriminant is zero, there is exactly one real root (a repeated root).
  • If the discriminant is negative, there are no real roots (the roots are complex numbers).

The Quadratic Formula is derived from completing the square, a method of solving quadratic equations by rewriting them in a perfect square form.

How to Use the Quadratic Formula

Using the Quadratic Formula to solve a quadratic equation involves several steps:

  1. Identify the coefficients a, b, and c in the quadratic equation ax² + bx + c = 0.
  2. Calculate the discriminant: D = b² - 4ac.
  3. If the discriminant is negative, the equation has no real roots. If it's zero, there's one real root. If it's positive, there are two real roots.
  4. Apply the Quadratic Formula: x = [-b ± √D] / (2a).
  5. Simplify the expression to find the roots.

Remember that when you take the square root of the discriminant, you must consider both the positive and negative roots (hence the ± symbol).

Let's look at an example to see how this works in practice.

Quadratic Formula Examples

Let's solve the quadratic equation x² - 5x + 6 = 0 using the Quadratic Formula.

  1. Identify the coefficients: a = 1, b = -5, c = 6.
  2. Calculate the discriminant: D = (-5)² - 4(1)(6) = 25 - 24 = 1.
  3. Since the discriminant is positive, there are two real roots.
  4. Apply the Quadratic Formula:

    x = [5 ± √1] / 2

  5. Calculate the two roots:
    • x₁ = (5 + 1)/2 = 6/2 = 3
    • x₂ = (5 - 1)/2 = 4/2 = 2

The roots of the equation x² - 5x + 6 = 0 are x = 2 and x = 3.

You can verify these roots by substituting them back into the original 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.

Here's another example with a negative discriminant: x² + 2x + 5 = 0.

  1. Identify the coefficients: a = 1, b = 2, c = 5.
  2. Calculate the discriminant: D = (2)² - 4(1)(5) = 4 - 20 = -16.
  3. Since the discriminant is negative, there are no real roots. The roots are complex numbers.
  4. Apply the Quadratic Formula:

    x = [-2 ± √(-16)] / 2 = [-2 ± 4i] / 2

  5. Simplify the roots:
    • x₁ = (-2 + 4i)/2 = -1 + 2i
    • x₂ = (-2 - 4i)/2 = -1 - 2i

Quadratic Formula FAQ

What is the Quadratic Formula used for?
The Quadratic Formula is used to find the roots of any quadratic equation. It's a reliable method that works for all quadratic equations, regardless of their coefficients.
Can the Quadratic Formula be used for non-quadratic equations?
No, the Quadratic Formula is specifically designed for quadratic equations (degree 2). For equations of higher or lower degree, different methods must be used.
What does the discriminant tell us about the roots?
The discriminant (b² - 4ac) provides information about the nature of the roots:
  • Positive discriminant: Two distinct real roots
  • Zero discriminant: One real root (a repeated root)
  • Negative discriminant: No real roots (complex roots)
Is the Quadratic Formula always accurate?
Yes, the Quadratic Formula is mathematically proven to be accurate for all quadratic equations where a ≠ 0. It provides exact solutions when the discriminant is a perfect square.
Can the Quadratic Formula be used to graph quadratic functions?
Yes, the roots found using the Quadratic Formula correspond to the x-intercepts of the quadratic function's graph. The vertex can be found using the formula x = -b/(2a).