To rewrite the quadratic equation [tex]\( y = -3x^2 - 18x - 25 \)[/tex] in the form [tex]\( y = a(x - p)^2 + q \)[/tex], we need to complete the square. Let's go through the process step-by-step:
1. Start with the given quadratic equation:
[tex]\[ y = -3x^2 - 18x - 25 \][/tex]
2. Factor out the coefficient of [tex]\(x^2\)[/tex] from the first two terms:
[tex]\[ y = -3(x^2 + 6x) - 25 \][/tex]
3. Complete the square inside the parentheses.
To do this, take the coefficient of [tex]\(x\)[/tex], which is 6, divide it by 2 to get 3, and then square it to get 9.
[tex]\[ x^2 + 6x = (x^2 + 6x + 9) - 9 \][/tex]
4. Incorporate this into the equation:
[tex]\[ y = -3((x^2 + 6x + 9) - 9) - 25 \][/tex]
5. Factor the perfect square trinomial:
[tex]\[ y = -3((x + 3)^2 - 9) - 25 \][/tex]
6. Distribute the [tex]\(-3\)[/tex] to both terms inside the parentheses:
[tex]\[ y = -3(x + 3)^2 + 27 - 25 \][/tex]
7. Simplify the constants:
[tex]\[ y = -3(x + 3)^2 + 2 \][/tex]
Thus, the quadratic equation [tex]\( y = -3x^2 - 18x - 25 \)[/tex] in the form [tex]\( y = a(x - p)^2 + q \)[/tex] is:
[tex]\[ y = -3(x + 3)^2 + 2 \][/tex]
So, the correct answer is:
[tex]\[ \boxed{y = -3(x+3)^2 + 2} \][/tex]
