Answer:
To complete the square for the given quadratic function \(h(x) = 4x^2 - 36x + 81\), follow these steps:
1. Factor out the coefficient of \(x^2\), which is 4:
\[h(x) = 4(x^2 - 9x) + 81\]
2. Take half of the coefficient of \(x\) (which is -9) and square it. Add and subtract this value inside the parentheses:
\[h(x) = 4(x^2 - 9x + \left(\frac{-9}{2}\right)^2 - \left(\frac{-9}{2}\right)^2) + 81\]
\[h(x) = 4(x^2 - 9x + \frac{81}{4} - \frac{81}{4}) + 81\]
3. Factor the trinomial inside the parentheses:
\[h(x) = 4\left((x - \frac{9}{2})^2 - \frac{81}{4}\right) + 81\]
4. Distribute the 4 and simplify:
\[h(x) = 4(x - \frac{9}{2})^2 - 81 + 81\]
\[h(x) = 4(x - \frac{9}{2})^2\]
Therefore, the completed square form is \(h(x) = 4(x - \frac{9}{2})^2\).
