Définition
Quadratic Equation
A quadratic equation is a second-order polynomial equation in a single variable x, with a non-zero coefficient for x². It has the standard form: ax² + bx + c = 0 where a, b, and c are constants.
Quadratic Formula
The quadratic formula is used to find the solutions (or roots) of a quadratic equation. The formula is: x = [-b ± √(b²-4ac)] / (2a).
Discriminant
In the quadratic formula, the expression under the square root sign, b²-4ac, is called the discriminant. It determines the nature of the roots.
Understanding the Quadratic Formula
The quadratic formula is a powerful tool in algebra, capable of providing solutions to any quadratic equation. By plugging the coefficients a, b, and c of a quadratic equation into the formula, one can find the values of x that satisfy the equation. The term '±' indicates that there are generally two solutions: one for adding the square root and one for subtracting it.
The Role of the Discriminant
The discriminant, given by b²-4ac, plays a crucial role in determining the nature of the roots of a quadratic equation. If the discriminant is positive, there are two distinct real roots. If it is zero, there is exactly one real root (a repeated root). If it is negative, the equation has two complex roots.
Solving Quadratic Equations Using the Formula
To solve a quadratic equation using the quadratic formula, first calculate the discriminant. Use it to determine the nature of the roots. Plug the values of a, b, and c into the quadratic formula and perform the arithmetic operations to find the values of x. Double-check your calculations to ensure accuracy.
Examples
Consider the quadratic equation 2x² + 3x - 2 = 0. Here, a = 2, b = 3, and c = -2. Calculate the discriminant: b² - 4ac = (3)² - 4(2)(-2) = 9 + 16 = 25. Since the discriminant is positive, the equation has two distinct real roots. Applying the quadratic formula: x = [-b ± √(b²-4ac)] / (2a) = [-3 ± √25] / 4. Therefore, the solutions are x = (-3 + 5) / 4 = 0.5 and x = (-3 - 5) / 4 = -2.
Now let's examine x² - 4x + 4 = 0. Here, a = 1, b = -4, and c = 4. The discriminant is b² - 4ac = (-4)² - 4(1)(4) = 16 - 16 = 0. Since the discriminant is zero, there is exactly one real root. Using the quadratic formula gives x = [4 ± √0] / 2 = 2. The solution is x = 2, which is a repeated root.
A retenir :
The quadratic formula is an essential tool for finding the roots of any quadratic equation. By understanding and applying the formula, students can solve quadratic equations of the form ax² + bx + c = 0. The discriminant, b²-4ac, helps determine the number and type of solutions: two real solutions if positive, one real solution if zero, and two complex solutions if negative. Practice with different quadratic equations enhances comprehension and proficiency in using the quadratic formula.
