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How Do You Calculate the Quadratic Formula?

Published in Quadratic Equations 3 mins read

The quadratic formula is a powerful tool used to solve any quadratic equation. It provides the solutions (roots) for x in equations of the form ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to zero.

Understanding the Formula

The quadratic formula itself is:

x = (-b ± √(b² - 4ac)) / 2a

Let's break down what each part means:

  • a, b, c: These are the coefficients from your quadratic equation. Make sure your equation is in standard form (ax² + bx + c = 0) before identifying these values.
  • -b: The negative of the coefficient b.
  • ±: This symbol means "plus or minus," indicating that there are usually two solutions for x.
  • √(b² - 4ac): This is the discriminant. It's the square root of (b squared minus four times a times c). The discriminant tells us about the nature of the roots (real, distinct, equal, or complex).
  • 2a: Two times the coefficient a.

Step-by-Step Calculation

Here's how to use the quadratic formula:

  1. Write the equation in standard form: Ensure your quadratic equation is in the form ax² + bx + c = 0.
  2. Identify a, b, and c: Determine the values of a, b, and c from your equation.
  3. Substitute into the formula: Plug the values of a, b, and c into the quadratic formula.
  4. Simplify: Perform the calculations carefully, following the order of operations (PEMDAS/BODMAS).
  5. Solve for x: Calculate the two solutions for x using the plus (+) and minus (-) signs separately.

Example

Let's solve the quadratic equation: 2x² + 5x - 3 = 0

  1. Standard form: The equation is already in standard form.
  2. Identify a, b, c: a = 2, b = 5, c = -3
  3. Substitute: x = (-5 ± √(5² - 4 * 2 * -3)) / (2 * 2)
  4. Simplify: x = (-5 ± √(25 + 24)) / 4 = (-5 ± √49) / 4 = (-5 ± 7) / 4
  5. Solve:
    • x = (-5 + 7) / 4 = 2 / 4 = 0.5
    • x = (-5 - 7) / 4 = -12 / 4 = -3

Therefore, the solutions are x = 0.5 and x = -3.

Numerous online calculators, like the ones found on Calculatorsoup, Calculator.net, and MathPapa, can assist in these calculations and show the step-by-step process. Khan Academy also provides helpful resources on this topic (https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:quadratic-formula-a1/a/quadratic-formula-explained-article).

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