The vertex of the parabola described by the equation 3y=2x^2-16x+18 is (m,n). What is m+n?

Answer:

the aswer is 32

Step-by-step explanation:

The vertex of the parabola described by the equation 3y=2x^2-16x+18 is (m,n). the sum of m + n is 178/27.

To find the vertex of the parabola described by the equation 3y = 2x^2 - 16x + 18, we first need to put the equation in the standard form of a parabola, which is y = a(x - h)^2 + k, where (h, k) represents the vertex.

Let's begin by dividing the entire equation by 3 to simplify it:

y = (2/3)x^2 - (16/3)x + 6

Now, we need to complete the square to rewrite the equation in the form y = a(x - h)^2 + k:

y = (2/3)(x^2 - (8/3)x) + 6

Next, we need to find the value that completes the square for the expression inside the parentheses. To do that, we take half of the coefficient of x and square it:

(8/3) / 2 = 4/3

(4/3)^2 = 16/9

Now, we add and subtract 16/9 inside the parentheses:

y = (2/3)(x^2 - (8/3)x + 16/9 - 16/9) + 6

Now, we group the first three terms and factor the perfect square trinomial:

y = (2/3)((x - 4/3)^2 - 16/9) + 6

Next, distribute the (2/3):

y = (2/3)(x - 4/3)^2 - (2/3)(16/9) + 6

Now, simplify:

y = (2/3)(x - 4/3)^2 - (32/27) + 6

Finally, bring everything together:

y = (2/3)(x - 4/3)^2 + (142/27)

Now we can see that the vertex of the parabola is at the point (h, k) = (4/3, 142/27).

To find m + n, we simply add the x-coordinate (h) and the y-coordinate (k):

m + n = 4/3 + 142/27

To add these fractions, we need to find a common denominator, which is 27. Convert 4/3 to have a denominator of 27:

4/3 = (4/3) * (9/9) = 36/27

Now we can add the fractions:

m + n = 36/27 + 142/27

m + n = (36 + 142)/27

m + n = 178/27

To know  more about  vertex of the parabola:

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