The Formula a(x - x1)(x - x2)
The formula a(x - x1)(x - x2) is a fundamental expression in algebra that represents a quadratic equation in factored form. This formula provides a concise and powerful way to express a parabola's equation, particularly when you know the x-intercepts and a point on the parabola.
Understanding the Parts
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a: Represents the coefficient of the quadratic term (x²). It influences the shape and direction of the parabola.
- If a is positive, the parabola opens upwards.
- If a is negative, the parabola opens downwards.
- The absolute value of a affects the width of the parabola. A larger absolute value makes the parabola narrower.
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x1 and x2: These are the x-coordinates of the points where the parabola intersects the x-axis, also known as the roots or zeros of the quadratic equation.
Advantages of Using the Formula
- Ease of Identifying Roots: The factored form directly reveals the roots of the equation.
- Direct Relationship to x-Intercepts: The formula clearly shows how the x-intercepts are incorporated into the equation.
- Simple Construction: It's straightforward to build the quadratic equation from the roots and a single point on the parabola.
Example
Let's say we know a parabola crosses the x-axis at x = 2 and x = -3, and it passes through the point (0, 6).
- Identify x1 and x2: x1 = 2 and x2 = -3
- Substitute into the formula: a(x - 2)(x + 3)
- Use the point (0, 6) to find a: 6 = a(0 - 2)(0 + 3) => a = -1
- Final Equation: -1(x - 2)(x + 3)
Therefore, the equation of the parabola is -1(x - 2)(x + 3).
Conclusion
The a(x - x1)(x - x2) formula offers a valuable tool for working with quadratic equations. It simplifies representing the equation, directly links the roots to the formula, and makes it easy to determine the equation given specific information about the parabola. This formula plays a significant role in understanding the behavior of parabolas and their connections to the x-intercepts.
