Question

Completing the square: 2x^2+8x-38=4

Answer 1

To complete the square, first you need to move the -38 to the right side. Can you do that?

Answer 2

Yes

Answer 3

Can you show what you got?

Answer 4

I got 2x^2+8x=42

Answer 5

Great.
The next step is that you must have the x^2 term with a coefficient of 1, but it is 2, so you need to divide both sides of the equation by 2. Can you do that and show?

Answer 6

Is it x^2+4x=21?

Answer 7

Great again.
Here is where the completing the square term comes in.
Now we rewrite the equation you wrote above, and we leave some room before the equal sign. We'll add the term that completes the square there.
I'll show you:
x^2 + 4x = 21

Answer 8

To find the term that completes the square, take the coefficient of the x-term, in this case 4, divide it by 2 then square it. What do you get?

Answer 9

When I squared it, I got 4. Correct?

Answer 10

Excellent.
Now we need to add the term you just found to that blank space we crateed, but remember than when we deal with an equation, we must do the same thing to both sides of the equation. If we add 4 to one side of an equation, we must add 4 to the other side too.

Answer 11

This is the step that completes the square:

\(x^2 + 4x \color{red}{+ 4} = 21 \color{red}{+ 4} \)

Answer 12

Now we can simplify the right side:

\(x^2 + 4x + 4 = 25\)

Also, if we added a term to the left side to "complete the square," that means the left side is now the square of a binomial:

\((x + 2)^2 = 25\)

There you have your complete the square procedure "completed."

Answer 13

Okay so what do you do after the (x+2)^2= 25?

Answer 14

If you need to solve the equation by completing the square, you need to use this technique:

For variable \(x\), and non-negative number, \(k\),
if \(x^2 = k\), then \(x = \pm\sqrt{k} \).

Answer 15

My final answer was x= -2 (plus or minus) 5. Is that correct?

Answer 16

Now we apply that technique to your problem.

\((x + 2)^2 = 25\)

We get

\(x + 2 = \pm\sqrt{25} \)

\( x + 2 = \pm5\)

\(x + 2 = 5\) or \(x + 2 = -5\)

\(x = 3\) or \(x = -7\)

Answer 17

Oh okay. I see. So the finaI answer would be x=3 and x=7?

Answer 18

3 and -7, not 7