To rewrite the quadratic equation [tex]\(y = 9x^2 + 9x - 1\)[/tex] in vertex form, we will complete the square. The vertex form of a quadratic equation is given by:
[tex]\[ y = a(x - h)^2 + k \][/tex]
Step-by-step, we can follow these instructions:
1. Identify the coefficients:
For the equation [tex]\(y = 9x^2 + 9x - 1\)[/tex], the coefficient [tex]\(a = 9\)[/tex], [tex]\(b = 9\)[/tex], and [tex]\(c = -1\)[/tex].
2. Factor out the coefficient of [tex]\(x^2\)[/tex] from the first two terms:
[tex]\[ y = 9(x^2 + x) - 1 \][/tex]
3. Complete the square:
- Find the value to complete the square inside the parentheses. Take half of the coefficient of [tex]\(x\)[/tex] (which is 1), square it, and add/subtract it inside the parentheses.
[tex]\[ \left(\frac{1}{2}\right)^2 = \frac{1}{4} \][/tex]
- Add and subtract this value inside the parentheses:
[tex]\[ y = 9(x^2 + x + \frac{1}{4} - \frac{1}{4}) - 1 \][/tex]
[tex]\[ y = 9((x + \frac{1}{2})^2 - \frac{1}{4}) - 1 \][/tex]
4. Distribute and simplify:
- Distribute the 9:
[tex]\[ y = 9(x + \frac{1}{2})^2 - 9 \cdot \frac{1}{4} - 1 \][/tex]
- Simplify the expression:
[tex]\[ y = 9(x + \frac{1}{2})^2 - \frac{9}{4} - 1 \][/tex]
- Combine the constants:
[tex]\[ y = 9(x + \frac{1}{2})^2 - \frac{9}{4} - \frac{4}{4} \][/tex]
[tex]\[ y = 9(x + \frac{1}{2})^2 - \frac{13}{4} \][/tex]
Therefore, the equation [tex]\(y = 9x^2 + 9x - 1\)[/tex] rewritten in vertex form is:
[tex]\[ y = 9\left(x + \frac{1}{2}\right)^2 - \frac{13}{4} \][/tex]
The correct answer is:
[tex]\[ \boxed{y=9\left(x+\frac{1}{2}\right)^2-\frac{13}{4}} \][/tex]
