What is the Quadratic Formula?
The quadratic formula is a powerful tool used to solve quadratic equations, which are equations of the form ax² + bx + c = 0. In this equation, a, b, and c are constants, and a cannot be equal to zero.
The Formula
The quadratic formula is given by:
x = (-b ± √(b² - 4ac)) / (2a)
Step-by-Step Explanation
- Identify the coefficients: Look for the values of a, b, and c in your equation. For instance, in the equation 2x² + 4x - 6 = 0, a = 2, b = 4, and c = -6.
- Calculate the discriminant: This is the part under the square root in the formula, known as b² - 4ac. Using our earlier example:
- b² = 4² = 16
- 4ac = 4 × 2 × (-6) = -48
- So, the discriminant is 16 - (-48) = 16 + 48 = 64.
- Substitute values into the formula: Now plug the b value and your discriminant into the quadratic formula:
- x = (-4 ± √64) / (2 × 2)
- Simplify: Calculate the square root of your discriminant and simplify:
- √64 = 8
- Now substitute back: x = (-4 ± 8) / 4
- Find the two possible solutions: Since you have the ± sign, you need to calculate both possibilities:
- First solution: x = (-4 + 8) / 4 = 4 / 4 = 1
- Second solution: x = (-4 - 8) / 4 = -12 / 4 = -3
- Conclusion: The solutions to the equation 2x² + 4x - 6 = 0 are x = 1 and x = -3.
Final Thoughts
Using the quadratic formula might seem complicated at first, but once you understand the steps, it becomes quite simple. Remember to practice with different equations to build your confidence!