The roots of a quadratic equation, specifically in the form ax² + bx + c = 0, are determined using the quadratic formula.
Understanding the Quadratic Formula
The quadratic formula provides a direct method for calculating the roots (also known as solutions or zeros) of a quadratic equation.
The Formula
The quadratic formula is:
x = (-b ± √ (b² - 4ac) ) / 2a
Where:
- 'a', 'b', and 'c' are the coefficients of the quadratic equation ax² + bx + c = 0.
- 'x' represents the roots (solutions) of the equation.
Discriminant Analysis
The discriminant, which is part of the quadratic formula, is denoted by D and is calculated as:
D = b² - 4ac
The discriminant helps determine the nature of the roots:
| Discriminant (D) | Nature of Roots |
|---|---|
| D > 0 | Two distinct and real roots |
| D = 0 | One real (repeated) root |
| D < 0 | Two complex (non-real) roots |
Example
Let's consider the equation 2x² + 5x + 2 = 0. Here, a=2, b=5 and c=2.
-
Calculate the discriminant.
D = b² - 4ac = (5)² - 4(2)(2) = 25-16=9.
Since D > 0, there will be two distinct real roots. -
Apply the quadratic formula:
x = (-b ± √ (b² - 4ac) ) / 2a
x = (-5 ± √9) / 2*2
x = (-5 ± 3) / 4
Thus,
x1 = (-5 + 3) / 4 = -2/4 = -1/2
x2 = (-5 - 3) / 4 = -8/4 = -2
The roots for 2x² + 5x + 2 = 0 are -1/2 and -2.
Step-by-Step Calculation:
- Identify Coefficients: From the given quadratic equation, determine the values of a, b, and c.
- Calculate the Discriminant: Compute the value of the discriminant, D = b² - 4ac. This will tell you the type of roots to expect.
- Apply the Formula: Substitute the values of a, b, and c into the quadratic formula.
- Solve for x: Simplify the expression to find the two roots of the equation. When D>0, take the plus and minus signs individually to calculate two distinct roots.
Practical Applications
The quadratic formula is essential in various fields, including:
- Physics: Calculating projectile motion and energy levels.
- Engineering: Solving problems in electrical circuits and structural analysis.
- Mathematics: Finding the intersection points of parabolas and lines.
By using the quadratic formula, one can effectively find the roots of any quadratic equation. The discriminant helps understand the nature of the solutions, whether real or complex, and how many distinct solutions exist.
