Quadratic formula
Based on the sources and our conversation, the quadratic formula is a powerful tool used to find the roots (or solutions) of a quadratic equation.
A quadratic equation is formed when a quadratic function, which is in the form ax² + bx + c where a is not equal to 0, is set equal to a value, often 0 for the standard form: ax² + bx + c = 0. The roots of this equation are the x-values for which the equation holds true. These roots correspond to the x-intercepts or zeros of the associated quadratic function f(x) = ax² + bx + c.
The quadratic formula provides a direct way to calculate these roots. It is given by the expression: x = [-b ± √(b² - 4ac)] / 2a
This formula is particularly significant because it is derived from the method of completing the square applied to the general quadratic equation ax² + bx + c = 0. The derivation involves steps like dividing by ‘a’ (assuming a is not 0), moving the constant term to the other side, adding (b/2a)² to both sides to complete the square, and then taking the square root of both sides.
Within the quadratic formula, the term b² - 4ac is known as the discriminant. Its value is crucial as it discriminates between the types and number of real roots the quadratic equation has. The rules for the discriminant are:
- If b² - 4ac > 0, there are two real roots. If the discriminant is also a perfect square, these two real roots are rational. If it is not a perfect square, the two real roots are irrational. This scenario corresponds graphically to the parabola intersecting the x-axis at two distinct points.
- If b² - 4ac = 0, there is one real root. This root is a repeated rational root. Graphically, this means the parabola touches the x-axis at exactly one point, which is its vertex.
- If b² - 4ac < 0, there are no real roots. This is because the square root of a negative number is not defined in the real number system, requiring the use of complex numbers which are not covered in this course. Graphically, the parabola does not intersect the x-axis at all.
The quadratic formula always works for finding the roots of a quadratic equation. It can be used regardless of whether the coefficients a, b, and c are rational or irrational numbers. The conditions on a, b, and c being rational are specifically needed if you want to distinguish between rational and irrational roots based on the discriminant.
Examples provided show the application of the formula to find roots. For instance, for x² + 2x - 24 = 0, with a=1, b=2, c=-24, the discriminant is 2² - 4(1)(-24) = 4 + 96 = 100. Since 100 > 0 and is a perfect square, there are two real rational roots. The formula gives x = [-2 ± √100] / 2(1) = [-2 ± 10] / 2, resulting in x = 4 or x = -6.
In summary, the quadratic formula is a universal method for solving quadratic equations, directly derived from completing the square, and the discriminant within it provides key information about the nature of the roots without needing to fully solve the equation.
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