Hello!
The first question is looking for the vertex form of a quadratic. From the vertex form, you can find the extreme value of the equation, or the vertex.
Vertex form is defined as y = a(x - h)² + k, where (h, k) is the vertex.
To convert to vertex form, first use the distributive property to take out the 3 from the first two terms.
y = 3x² + 30x + 71
y = 3(x² + 10x) + 71
Now, complete the square for the equation inside the parenthesis. This can be done by seeing which number makes a perfect binomial. Remember that you must multiply it by 3 as there is the 3 outside the parenthesis.
y + 75 = 3(x² + 10x) + 75 + 71
y + 75 = 3(x² + 10x) + 3(25) + 71
y + 75 = 3(x² + 10x + 25) + 71
y = 3(x² + 10x + 25) - 4
And then, turn the equation into a perfect square binomial.
y = 3(x + 5)² - 4
Now you have your vertex form.
Looking at the values, -5 is h and -4 is k (as recall in the original equation, you are subtracting h and adding k).
Therefore, the vertex is (-5, -4).
Hope this helped!
