SOLUTION: D..ex I have no idea on how to do these types of problems Problem #7 Solve by completing the square x^2=5x+2 Problem #8 Find the x-intercepts y= x^2+5x+2 Problem #9

Question 78077: D..ex
I have no idea on how to do these types of problems
Problem #7
Solve by completing the square
x^2=5x+2
Problem #8
Find the x-intercepts
y= x^2+5x+2
Problem #9
Is this a trinomial a perfect square? Explain why or why not.
x^2+18x+81

Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
#7

Get all terms to one side

Solved by pluggable solver: Completing the Square to Get a Quadratic into Vertex Form

Start with the given equation

Add to both sides

Factor out the leading coefficient

Take half of the x coefficient to get (ie ).

Now square to get (ie )

Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of does not change the equation

Now factor to get

Distribute

Multiply

Now add to both sides to isolate y

Combine like terms

Now the quadratic is in vertex form where , , and . Remember (h,k) is the vertex and "a" is the stretch/compression factor.

Check:

Notice if we graph the original equation we get:

Graph of . Notice how the vertex is (,).

Notice if we graph the final equation we get:

Graph of . Notice how the vertex is also (,).

So if these two equations were graphed on the same coordinate plane, one would overlap another perfectly. So this visually verifies our answer.

So we get:

Add 8.25 to both sides
Take the square root of both sides
Add 2.5 to both sides

So we get:
or
Which is approximately
or

#8
To find the x intercepts we can use the quadratic formula:

Solved by pluggable solver: SOLVE quadratic equation with variable

Quadratic equation (in our case ) has the following solutons:

For these solutions to exist, the discriminant should not be a negative number.

First, we need to compute the discriminant : .

Discriminant d=17 is greater than zero. That means that there are two solutions: .

Quadratic expression can be factored:

Again, the answer is: -0.43844718719117, -4.56155281280883. Here's your graph:

So our x-intercepts are
and

#9
The trinomial is a perfect square since it can be written as

Which is a representation of a squared number. For instance if we let x=1 we get

And we can see that 100 is a perfect square. Notice we can let x=1 for this equation:
which gives us the same result.
So since the expression generates perfect squares for any whole x's (make a table and you'll see it), it is a perfect square
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