- Identify the axis of symmetry using the formula \(x = -b / (2a)\), which results in \(x = 1\).
- Set the equation \(-3x^2 + 6x - 1 = 1\) to standard form: \(3x^2 - 6x + 2 = 0\).
- Apply the quadratic formula to find the roots: \(x = \frac{6 \pm \sqrt{12}}{6}\).
- Simplify the roots to obtain the final values: \(x = 1 - \frac{\sqrt{3}}{3}\) and \(x = 1 + \frac{\sqrt{3}}{3}\).
- The final solutions are \(x = \boxed{1 - \frac{\sqrt{3}}{3}, 1 + \frac{\sqrt{3}}{3}}\).
Explanation
- Analyzing the Quadratic Function and Finding the Axis of Symmetry
To analyze the quadratic function \(y = -3x^2 + 6x - 1\), we first identify its key features. The function is in the form \(y = ax^2 + bx + c\), where \(a = -3\), \(b = 6\), and \(c = -1\). The vertex of the parabola is given as \((1, 2)\). The axis of symmetry for a quadratic function \(y = ax^2 + bx + c\) is a vertical line passing through the x-coordinate of the vertex, given by the formula \(x = -b / (2a)\). Substituting our values, we get \(x = -6 / (2 \times -3) = -6 / -6 = 1\). Therefore, the equation of the line of symmetry is \(x = 1\).
- Solving the Equation Algebraically
We are asked to solve the equation \(-3x^2 + 6x - 1 = 1\). Geometrically, this represents finding the x-coordinates where the parabola \(y = -3x^2 + 6x - 1\) intersects the horizontal line \(y = 1\). To solve this algebraically, we move all terms to one side of the equation:
\[-3x^2 + 6x - 1 - 1 = 0\]
\[-3x^2 + 6x - 2 = 0\]
Multiplying by \(-1\) for convenience, we have the standard form:
\[3x^2 - 6x + 2 = 0\]
Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) with \(a = 3\), \(b = -6\), and \(c = 2\):
\[x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(3)(2)}}{2(3)}\]
\[x = \frac{6 \pm \sqrt{36 - 24}}{6}\]
\[x = \frac{6 \pm \sqrt{12}}{6} = \frac{6 \pm 2\sqrt{3}}{6} = 1 \pm \frac{\sqrt{3}}{3}\]
Thus, the two solutions are \(x = 1 + \frac{\sqrt{3}}{3}\) and \(x = 1 - \frac{\sqrt{3}}{3}\).
- Conclusion and Final Answer
The two solutions obtained are \(x = 1 + \frac{\sqrt{3}}{3} \approx 1.577\) and \(x = 1 - \frac{\sqrt{3}}{3} \approx 0.423\). These represent the x-coordinates where the parabola crosses the line \(y = 1\). The line of symmetry \(x = 1\) serves as the midpoint between these two solutions, illustrating the symmetry of the parabola. The final solutions for the equation are \(x = 1 - \frac{\sqrt{3}}{3}\) and \(x = 1 + \frac{\sqrt{3}}{3}\).
Examples
In engineering, such quadratic equations often model the path of a projectile. If a bridge needs to be built at a certain height, the intersection points calculated above would represent the specific locations on the ground (horizontal distance) where the trajectory of a launched object (like a ball or rocket) reaches that specific height. Understanding how to find these intersection points is vital for planning trajectories and ensuring structural safety in architectural design.