Simplifying equation after completing the square

[pencile]

[FONT=MJXc-TeX-main-R, MJXc-TeX-main-Rw]1 - 4x - 4x²
= -4 (x² + x - 1/4)
= -4 (x² + x + 1/4 - 1/4 - 1/4)
= -4 [(x + 1/2)² - 1/2]

How do you simplify [/FONT]-4 [(x + 1/2)² - 1/2] to 2 - (2x + 1)² as in the following?[FONT=MJXc-TeX-main-R, MJXc-TeX-main-Rw]
[/FONT]

Problem:
∫

1​

√−4x2−4x+1​

dx∫1−4x2−4x+1dx
Complete the square:
=∫

1​

√2−(2x+1)2​

dx​

 

[mmm4444bot]

1 - 4x - 4x²
= -4 (x² + x - 1/4)
= -4 (x² + x + 1/4 - 1/4 - 1/4)
= -4 [(x + 1/2)² - 1/2]


How do you simplify -4 [(x + 1/2)² - 1/2] to 2 - (2x + 1)² …

First, look at the expanded form of (2x+1)^2, so you know what you're aiming for.

Second, try distributing the -4.

Do you then see a substitution? :cool:

following?

Problem:
∫

1
√−4x2−4x+1​

dx∫1−4x2−4x+1dx
Complete the square:
=∫

1
√2−(2x+1)2​

dx​

As you can see, your attempt to "draw" ratios failed. If you desire to draw stuff with text in this forum, you need to use the code tags (to retain repeated spaces). Also, the fixed-width font Courier New is best for alignment while composing.

Otherwise, you could use the link in the forum guidelines (which you read before posting), to learn how to type ratios, exponents and radicals as text.

Then, you can type integrals similar to the following.

Indefinite integral:

int(1/sqrt[2 - (2x + 1)^2] dx)

or

∫ 1/√[2 - (2x + 1)^2] dx


Definite integral:

int([a..b] 1/sqrt[2 - (2x + 1)^2] dx)

or

∫ [a..b] 1/√[2 - (2x + 1)^2] dx


:idea: In the future, you can use the Preview Post button, to see how your post will render, for proofreading before submission.

 

[stapel]

1 - 4x - 4x²
= -4 (x² + x - 1/4)
= -4 (x² + x + 1/4 - 1/4 - 1/4)
= -4 [(x + 1/2)² - 1/2]

How do you simplify -4 [(x + 1/2)² - 1/2] to 2 - (2x + 1)² as in the following?

Problem:
∫

1
√−4x2−4x+1​

dx∫1−4x2−4x+1dx
Complete the square:
=∫

1
√2−(2x+1)2​

dx​

​

I have no idea what is going on in the second half of your post, so I'll just respond to the first half:

---

How do you simplify -4 [(x + 1/2)² - 1/2] to 2 - (2x + 1)²...?

---

You know that 4 = 2², right? So 4(x^2) = (2x)^2, and 4(1/2)^2 = 4(1/4) = 1. Also, you know that multiplying a minus by a minus results in a plus, right? So -4(-(1/4)) = +1. And multiplying a minus by a plus results in a minus, so -4(x^2) = -4x^2 = -(2x)^2.

So please reply with specifications regarding the point at which the working stops making sense to you. Thank you!