x^2+6x=16 Solve For X by Completing the Square... Show work step by step

Question

Thank You SOOO MUCH... I am struggling really badly in Math right now and don't have the money to pay for a tutor, so I appreciate this incredibley.

Answer 1

Hi Tae!

Completing the square can be tricky, I remember I had a tough time learning it too.

One phrase that has helped me is "half it, square it, add it to both sides."

What is "It," though?

Let's first take a look at the basic formula for your equation:

ax²+bx+c

The phrase applies to the "B" factor.

Let's look at your equation:

x²+6x=16

The B in this equation would be 6 because it is the middle factor.

So now, back to the phrase: "Half it, square it, add to both sides"

6/2=3

3²=9

x²+6x+9=16+9

The importance of this is: now, the left side can be put into factored form!

(x+3)²=25

*The three is the number that occurred when you halved the middle term.

Now, to solve, you can take the square root of both sides!

x+3=±√25

x+3=±5

Now, you isolate x

x=-3±5

-3+5=2

-3-5=-8

x=2 and -8

Hope this helped!

If you are still having trouble but you aren't financially able to get a tutor, I can give you a session free of charge

Answer 2

First think about examples of a perfect square and the mechanics of them. For example, if you had (x+1)² and you were to expand it, you would come up with x² + 2x +1. Similarly, if you had (x+5)² the expansion would be x² + 10x +25. Notice the pattern emerging here. The second coefficient is the square root of the last one doubled. In other words, in both cases, if you take the last number and take its square root and double that answer, you get the second coefficient. This is the key to understanding complete the square.

For this problem ask look at the first two terms only x² + 6x. What would the third number need to be keeping in mind the pattern I just described. It would need to be a 9 to factor to (x+3)². So just add a 9, but wait - you can't just add a 9 unless you do it to both sides of the equation. Here you go:

x² + 6x = 16 (original) - now add 9 to both sides:

x² + 6x + 9 = 16 +9 or

x² + 6x +9 = 25 or

(x² + 6x +9 - 25 = 0 or

(x+3)².- 25 = 0

Graphically, by transformation, this is the x² curve shifted to the left by 3 and up by 25.

Answer 3

Completing the square:

x² + 6x = 16

x² + 6x - 16 = 0

x² + 2 • (6/2) • x - 16 = 0

(x²) + 2 • (3) • (x) + (3²) - 3² - 16 = 0

(x² + 2 • 3 • x + 3²) - 3² - 16 = 0

(x + 3)² - 9 - 16 = 0

(x + 3)² - 25 = 0

(x + 3)² = 25