SOLUTION: 1. Solve {{{x^2+8x=11}}} by completing the square. Find exact solutions. Then: a. Find the vertex. b. Find the line of symmetry. c. Determine whether there is a maximum o

Question 143722: 1. Solve by completing the square. Find exact solutions.
Then:
a. Find the vertex.
b. Find the line of symmetry.
c. Determine whether there is a maximum or minimum value and find that value.
d. Show a sketch of the graph.

Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Start with the given equation

Take half of the x-coefficient 8 to get 4 (ie )

Now square 4 to get 16. (ie )

Add this result to both sides

Combine like terms

Factor to get . (note: if you need help with factoring, check out this solver)

Take the square root of both sides

Subtract 4 from both sides

Simplify to get . (note: If you need help with simplifying square roots, check out this solver)

So the solutions are

-----------------------------

a)

Start with the given equation

Subtract 11 from both sides

To find the vertex, we first need to find the axis of symmetry (ie the x-coordinate of the vertex)
To find the axis of symmetry, use this formula:

From the equation we can see that a=1 and b=8

Plug in b=8 and a=1

Multiply 2 and 1 to get 2

Reduce

So the axis of symmetry is

So the x-coordinate of the vertex is . Lets plug this into the equation to find the y-coordinate of the vertex.

Lets evaluate

Start with the given polynomial

Plug in

Raise -4 to the second power to get 16

Multiply 8 by -4 to get -32

Now combine like terms

So the vertex is (-4,-27)

b)

From part a) we found the axis of symmetry to be

c)

Looking at , we can see that , , and . Since , this tells us that the parabola opens upward and that there is a minimum.

To find the minimum, we only need to look at the vertex. Since the vertex is the point (-4,-27), this means that the minimum is

d)

Here's a sketch to visually verify our answers

Graph of
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