Rewrite in vertex form:
$-3 x^2+36 x-114$
We are given the quadratic expression \(-3x^2 + 36x - 114\). Our goal is to rewrite this expression in the vertex form \(a(x - h)^2 + k\), where \((h, k)\) is the vertex of the parabola. This form helps us understand the properties of the quadratic function, such as its maximum or minimum value and its axis of symmetry.
To begin, we factor the leading coefficient, \(a = -3\), out of the terms involving \(x\):
\[-3x^2 + 36x - 114 = -3(x^2 - 12x) - 114.\]
Now, we focus on the expression inside the parentheses, \(x^2 - 12x\). To complete the square, we need to add a constant \(d\) such that \(x^2 - 12x + d\) becomes a perfect square. The constant is the square of half the coefficient of \(x\), which is \(\left(\frac{-12}{2}\right)^2 = (-6)^2 = 36\).
We add and subtract 36 inside the parentheses to maintain the equality:
\[-3(x^2 - 12x + 36 - 36) - 114.\]
Next, we rewrite the perfect square trinomial \(x^2 - 12x + 36\) as \((x - 6)^2\):
\[-3((x - 6)^2 - 36) - 114.\]
Finally, distribute the \(-3\) to both terms inside the large parentheses:
\[-3(x - 6)^2 + (-3 \times -36) - 114 = -3(x - 6)^2 + 108 - 114.\]
Combine the constant terms: \(108 - 114 = -6\). The vertex form is \(-3(x - 6)^2 - 6\).
Vertex form is incredibly useful in physics and engineering. For example, if you are modeling the trajectory of a projectile launched into the air, the path is a parabola. The vertex of this parabola represents the maximum height reached by the object. By rewriting the equation in vertex form, you can immediately identify the maximum height and the time at which it occurs, simplifying the process of analyzing motion without having to re-solve the equation.