Solving quadratic equation sums involves finding the values of the variable (often 'x') that make the equation equal to zero. A quadratic equation is typically in the form ax² + bx + c = 0, where a, b, and c are coefficients and a ≠ 0. The most common method to solve these equations is by using the quadratic formula.
The Quadratic Formula
The quadratic formula is:
x = (-b ± √(b² - 4ac)) / 2a
This formula gives you two possible solutions for 'x', because of the ± sign.
Step-by-Step Guide to Using the Quadratic Formula
Here's how to solve quadratic equations using the quadratic formula, along with an example incorporating the reference:
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Identify a, b, and c: In the quadratic equation ax² + bx + c = 0, determine the values of a, b, and c.
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Plug into the Formula: Substitute the values of a, b, and c into the quadratic formula.
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Calculate the Discriminant: The expression inside the square root (b² - 4ac) is called the discriminant. Calculate this first. This part tells us about the nature of the roots.
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Solve for x: Complete the calculation to find the two values of x.
- Calculate the solution using the "+" sign from the ±.
- Then calculate using the "−" sign from the ±.
Example from the Reference
Let's say you have the quadratic equation where after completing steps 1 and 2, the equation looks like this:
x = (-3 ± √(5²)) / 4
According to the reference we then get two results, like so:
- First Solution: x = (-3 + 5) / 4 = 2/4 = 1/2
- Second Solution: x = (-3 - 5) / 4 = -8/4 = -2
These two values for x are the solutions to the quadratic equation.
Different Methods
While the quadratic formula is a reliable method, other methods can also be used depending on the specific equation:
- Factoring: If the quadratic equation can be factored, you can easily find the solutions by setting each factor equal to zero. This method is faster if factoring is obvious but not applicable to all equations.
- Completing the Square: This method involves rewriting the quadratic equation in a specific form to find solutions. It is a longer process but is particularly helpful in some specific cases or when needing to derive the quadratic formula itself.
Summary of Steps
| Step | Description |
|---|---|
| 1. Identify a, b, and c | Determine the coefficients of the quadratic equation ax² + bx + c = 0. |
| 2. Apply Quadratic Formula | Substitute a, b, and c values into the formula: x = (-b ± √(b² - 4ac)) / 2a. |
| 3. Calculate Discriminant | Compute b² - 4ac. This informs you about the nature of the solutions. |
| 4. Solve for x | Compute both solutions using + and - in the ±. This leads to x = (-b + √(b² - 4ac)) / 2a and x = (-b - √(b² - 4ac)) / 2a. |
By following these steps, you can effectively solve quadratic equation sums.
