Explanation
The quadratic formula is a method to calculate the roots of a quadratic function. A quadratic function has the general form f(x) = ax^2 + bx + c. The roots are the points where the parabola intersects the x-axis. The quadratic formula is:
The quadratic formula for calculating the roots is:
Procedure
To calculate the roots of a quadratic function using the quadratic formula, first determine the values of a, b, and c from the function’s equation. Then substitute these values into the quadratic formula and compute the solutions.
Examples
Calculate the roots of the quadratic function f(x) = x^2 - 5x + 6.
Here we have: a=1,\, b=-5,\, c=6. Substituting into the quadratic formula gives: x_{1,2} = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot 6}}{2 \cdot 1}. It follows that: x_1 = 3 and x_2 = 2.
Calculate the roots of the quadratic function f(x) = 2x^2 + 4x + 2.
Here we have: a=2,\, b=4,\, c=2. Substituting yields: x_{1,2} = \frac{-4 \pm \sqrt{16 - 16}}{4}. It follows that: x_1 = x_2 = -1. The parabola touches the x-axis at exactly one point.
Note
- ★The quadratic formula is used to calculate the roots of quadratic functions.
- ★The discriminant b^2 - 4ac determines whether there are two, one, or no real roots.
Exercises
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