Let's solve each part of the problem step-by-step.
### Part 7.5) Find the vertex equation by completing the square.
The given quadratic equation is:
[tex]\[ y = 3x^2 - 12x + 6 \][/tex]
To find the vertex, we will complete the square for this equation.
1. Factor out the coefficient of [tex]\( x^2 \)[/tex] from the first two terms:
[tex]\[ y = 3(x^2 - 4x) + 6 \][/tex]
2. Complete the square inside the parenthesis:
To complete the square, we need to add and subtract a term inside the parenthesis that makes the expression a perfect square trinomial. The term to add and subtract is [tex]\( \left(\frac{-4}{2}\right)^2 = 4 \)[/tex].
[tex]\[ y = 3(x^2 - 4x + 4 - 4) + 6 \][/tex]
[tex]\[ y = 3((x - 2)^2 - 4) + 6 \][/tex]
3. Simplify the equation:
Distribute the 3 and combine the constants:
[tex]\[ y = 3(x - 2)^2 - 12 + 6 \][/tex]
[tex]\[ y = 3(x - 2)^2 - 6 \][/tex]
The equation in vertex form is:
[tex]\[ y = 3(x - 2)^2 - 6 \][/tex]
From this, we identify the vertex [tex]\((h, k)\)[/tex]. The vertex form [tex]\( y = a(x - h)^2 + k \)[/tex] shows the vertex directly as the point [tex]\((h, k)\)[/tex].
So, the vertex of the parabola is:
[tex]\[ (h, k) = (2, -6) \][/tex]
Given the choices, the correct vertex is:
[tex]\[ (2, -6) \][/tex]
### Part 8) Subtract the complex numbers.
The given complex numbers are:
[tex]\[ z_1 = 3 - 2i \][/tex]
[tex]\[ z_2 = 7 + 6i \][/tex]
To find the result of subtracting [tex]\( z_2 \)[/tex] from [tex]\( z_1 \)[/tex]:
[tex]\[ (3 - 2i) - (7 + 6i) \][/tex]
Subtract the real parts and the imaginary parts separately:
Real part:
[tex]\[ 3 - 7 = -4 \][/tex]
Imaginary part:
[tex]\[ -2i - 6i = -8i \][/tex]
Therefore, the result of the subtraction is:
[tex]\[ (3 - 2i) - (7 + 6i) = -4 - 8i \][/tex]
The result is:
[tex]\[ -4 - 8i \][/tex]
So, we have successfully solved both parts of the problem.
