To find the roots of a quadratic equation, you can use the quadratic formula. A quadratic equation is in the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The quadratic formula provides a direct way to calculate the roots (also called solutions or zeros) of the equation.
The Quadratic Formula
The quadratic formula is:
x = (-b ± √ (b² – 4ac) )/2a
Where:
- x represents the roots of the quadratic equation.
- a, b, and c are the coefficients from the quadratic equation ax² + bx + c = 0.
Steps to find the roots:
-
Identify a, b, and c: Determine the values of a, b, and c from your quadratic equation.
-
Calculate the Discriminant: The discriminant (D) is the part of the formula under the square root:
- D = b² – 4ac
The discriminant tells you about the nature of the roots:
- If D > 0: Two distinct real roots.
- If D = 0: One real root (a repeated or double root).
- If D < 0: Two complex roots (no real roots).
-
Apply the Quadratic Formula: Plug the values of a, b, and c into the quadratic formula and simplify. This will give you two possible solutions for x (one using the plus sign and one using the minus sign).
Example
Let's consider the quadratic equation: 2x² + 5x - 3 = 0
-
Identify a, b, and c:
- a = 2
- b = 5
- c = -3
-
Calculate the Discriminant:
- D = (5)² – 4 2 (-3) = 25 + 24 = 49
-
Apply the Quadratic Formula:
- x = (-5 ± √49) / (2 * 2)
- x = (-5 ± 7) / 4
Therefore, the two roots are:
- x₁ = (-5 + 7) / 4 = 2 / 4 = 1/2
- x₂ = (-5 - 7) / 4 = -12 / 4 = -3
So, the roots of the equation 2x² + 5x - 3 = 0 are x = 1/2 and x = -3.
Summary Table
| Concept | Description | Formula |
|---|---|---|
| Quadratic Equation | An equation of the form ax² + bx + c = 0, where a ≠ 0. | ax² + bx + c = 0 |
| Roots | The solutions to the quadratic equation (the values of x that make the equation true). | |
| Quadratic Formula | The formula used to find the roots of a quadratic equation. | x = (-b ± √ (b² – 4ac) )/2a |
| Discriminant | The part of the quadratic formula under the square root, which determines the nature of the roots. | D = b² – 4ac |
