The quadratic formula is derived to solve a quadratic equation, not the other way around. You don't "turn" a quadratic equation into the quadratic formula. Instead, you use the quadratic formula to find the solutions (roots) of a quadratic equation. Here's how it works, utilizing information from the reference:
First, you need a quadratic equation in the standard form:
ax² + bx + c = 0
where a, b, and c are coefficients.
Then, you take these coefficients and plug them into the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
Understanding the Process
The quadratic formula provides the values of x that satisfy the quadratic equation. The "±" symbol indicates that there are generally two possible solutions:
- One solution using the plus sign: x = (-b + √(b² - 4ac)) / (2a)
- Another solution using the minus sign: x = (-b - √(b² - 4ac)) / (2a)
Example
Let's say we have the quadratic equation:
2x² + 5x - 3 = 0
Here, a = 2, b = 5, and c = -3. Plugging these into the quadratic formula:
x = (-5 ± √(5² - 4 2 -3)) / (2 * 2)
x = (-5 ± √(25 + 24)) / 4
x = (-5 ± √49) / 4
x = (-5 ± 7) / 4
So the two solutions are:
- x = (-5 + 7) / 4 = 2 / 4 = 1/2
- x = (-5 - 7) / 4 = -12 / 4 = -3
Therefore, the solutions to the equation 2x² + 5x - 3 = 0 are x = 1/2 and x = -3.
In summary:
| Step | Description |
|---|---|
| 1 | Get the quadratic equation into standard form: ax² + bx + c = 0 |
| 2 | Identify the coefficients a, b, and c. |
| 3 | Plug the coefficients into the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a) |
| 4 | Simplify and solve for x. |
