The sum of the roots of a quadratic equation can be expressed using a simple formula derived from the coefficients of the equation.
Understanding Quadratic Equations
A quadratic equation is a polynomial equation of the second degree, generally written as:
ax² + bx + c = 0
Where:
- 'a', 'b', and 'c' are coefficients, with 'a' not equal to 0.
- 'x' is the variable.
The roots of the equation are the values of 'x' that satisfy the equation.
Expressing the Sum of Roots
According to the reference, the sum of the roots of a quadratic equation is calculated as follows:
Sum of Roots = - (Coefficient of x) / (Coefficient of x²)
In terms of the quadratic equation ax² + bx + c = 0, this translates to:
Sum of Roots = -b / a
Example
Let's consider the quadratic equation:
2x² - 5x + 3 = 0
Here, a = 2, b = -5, and c = 3.
Using the formula:
Sum of Roots = - (-5) / 2 = 5 / 2
Therefore, the sum of the roots for this quadratic equation is 5/2.
Practical Application
This formula is incredibly useful because:
- It allows you to find the sum of the roots without actually calculating the roots individually.
- It is directly related to the coefficients of the quadratic equation.
- It aids in quickly checking if your roots are correct, especially when calculated using other methods.
- It also provides insight into the behaviour of quadratic equations without having to explicitly solve them.
Key Points
- The sum of roots of a quadratic equation can be readily obtained from the coefficients of x and x².
- The formula is: Sum of Roots = -b/a.
- This method simplifies root analysis without explicitly solving the equation.
- The reference states the sum of roots is “the negative of the coefficient of x divided by the coefficient of x^2”.
| Feature | Description | Formula |
|---|---|---|
| Sum of Roots | The sum of the two solutions (roots) of the quadratic equation. | -b/a |
