how to do quadratic formula

Daniel Parker

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10-03-2025

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The quadratic formula is a powerful tool for solving quadratic equations—equations of the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. While factoring can solve some quadratic equations, the quadratic formula works for all quadratic equations, even those that are difficult or impossible to factor. This guide will walk you through using the quadratic formula step-by-step, with examples.

Understanding the Quadratic Formula

The quadratic formula itself is:

x = [-b ± √(b² - 4ac)] / 2a

Let's break down what each part means:

• x: Represents the solutions (or roots) of the quadratic equation. A quadratic equation can have two, one, or zero real solutions.
• a, b, c: These are the coefficients from your quadratic equation (ax² + bx + c = 0). Make sure your equation is in standard form before you start.
• ±: This symbol means "plus or minus," indicating that there are usually two possible solutions for x. One solution uses the "+" and the other uses the "-".
• √: This is the square root symbol.
• b² - 4ac: This part is called the discriminant. It tells you about the nature of the solutions:
  • b² - 4ac > 0: Two distinct real solutions.
  • b² - 4ac = 0: One real solution (a repeated root).
  • b² - 4ac < 0: No real solutions (two complex solutions involving imaginary numbers).

Step-by-Step Guide to Using the Quadratic Formula

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

Step 1: Identify a, b, and c.

In our equation, 2x² + 5x - 3 = 0:

• a = 2
• b = 5
• c = -3

Step 2: Substitute the values into the quadratic formula.

x = [-5 ± √(5² - 4 * 2 * -3)] / (2 * 2)

Step 3: Simplify the expression under the square root (the discriminant).

x = [-5 ± √(25 + 24)] / 4 x = [-5 ± √49] / 4

Step 4: Calculate the square root.

√49 = 7

Step 5: Solve for the two possible values of x.

• Using the "+": x = (-5 + 7) / 4 = 2 / 4 = 0.5
• Using the "-": x = (-5 - 7) / 4 = -12 / 4 = -3

Therefore, the solutions to the quadratic equation 2x² + 5x - 3 = 0 are x = 0.5 and x = -3.

Example with a Discriminant Less Than Zero

Let's consider the equation x² + x + 1 = 0.

• a = 1
• b = 1
• c = 1

The discriminant is b² - 4ac = 1² - 4 * 1 * 1 = -3. Since the discriminant is negative, there are no real solutions. The solutions would involve imaginary numbers (involving 'i', where i² = -1).

How to Check Your Answers

You can always check your answers by substituting them back into the original quadratic equation. If the equation holds true, your solutions are correct.

Frequently Asked Questions (FAQs)

Q: What if the equation isn't in standard form?

A: Rearrange the equation so it's in the form ax² + bx + c = 0 before identifying a, b, and c.

Q: What if 'a' is 0?

A: The quadratic formula doesn't apply if a = 0. The equation is then linear, not quadratic, and can be solved using simpler algebraic methods.

Q: Can I use a calculator?

A: Absolutely! Calculators are helpful for simplifying the more complex parts of the quadratic formula, especially the discriminant.

By following these steps and understanding the concepts involved, you'll master the quadratic formula and be able to solve a wide range of quadratic equations. Remember to always double-check your work!