Understanding Quadratic Equations
Quadratic equations are fundamental in mathematics, often appearing in various applications ranging from physics to finance. A quadratic equation is typically expressed in the standard form:
- ax² + bx + c = 0
where:
- a, b, and c are constants, and
- a ≠ 0.
Solving Quadratic Equations
There are several methods to solve quadratic equations:
- Factoring
- Completing the square
- Quadratic formula (x = (-b ± √(b² - 4ac)) / 2a)
Let’s delve into the quadratic formula method, as it is widely applicable.
Step-by-Step Solution Using the Quadratic Formula
1. Identify Coefficients:
- From the equation ax² + bx + c = 0, identify a, b, and c.
2. Calculate the Discriminant:
- Discriminant (D) = b² - 4ac.
- This value determines the nature of the roots:
- If D > 0: Two distinct real roots.
- If D = 0: One real root (repeated).
- If D < 0:="" no="" real="" roots="" (complex="" />
3. Apply the Quadratic Formula:
- Substitute the values of a, b, and D into the formula:
- x = (-b ± √D) / (2a).
- This will yield the solutions for the variable x.
Example Problem
Consider the equation 2x² - 4x - 6 = 0.
- Here, a = 2, b = -4, c = -6.
- Calculate D: D = (-4)² - 4 * 2 * (-6) = 16 + 48 = 64 (D > 0).
- Apply the formula:
- x = (4 ± √64) / (4) = (4 ± 8) / 4.
- Roots: x = 3 and x = -1.
This method provides a clear, systematic approach to solving quadratic equations effectively.