put into vertex form
1. $y = x^2 - 12x - 25$
3. $y = x^2 + 2x - 3$
2. $y = x^2 + 4x + 5$
4. $y = x^2 - 20x + 101$
put into vertex form
5. $y = 2x^2 + 8x + 1$
6. $y = -3x^2 + 18x + 5$

Question

Accepted Answer

### Problem 1: $ y = x^2 - 12x - 25 $
# Explanation:
## Step 1: Complete the square for $ x^2 - 12x $
Take the coefficient of $ x $, which is -12. Half of -12 is -6, and squaring -6 gives 36. So we add and subtract 36 inside the equation.
$ y = (x^2 - 12x + 36) - 36 - 25 $
## Step 2: Factor the perfect square trinomial
The expression $ x^2 - 12x + 36 $ factors into $ (x - 6)^2 $. Then simplify the constants.
$ y = (x - 6)^2 - 61 $
# Answer:
$ y = (x - 6)^2 - 61 $

### Problem 2: $ y = x^2 + 4x + 5 $
# Explanation:
## Step 1: Complete the square for $ x^2 + 4x $
The coefficient of $ x $ is 4. Half of 4 is 2, and squaring 2 gives 4. Add and subtract 4.
$ y = (x^2 + 4x + 4) - 4 + 5 $
## Step 2: Factor and simplify
$ x^2 + 4x + 4 $ factors to $ (x + 2)^2 $, and -4 + 5 = 1.
$ y = (x + 2)^2 + 1 $
# Answer:
$ y = (x + 2)^2 + 1 $

### Problem 3: $ y = x^2 + 2x - 3 $
# Explanation:
## Step 1: Complete the square for $ x^2 + 2x $
Coefficient of $ x $ is 2. Half of 2 is 1, squaring 1 gives 1. Add and subtract 1.
$ y = (x^2 + 2x + 1) - 1 - 3 $
## Step 2: Factor and simplify
$ x^2 + 2x + 1 = (x + 1)^2 $, and -1 - 3 = -4.
$ y = (x + 1)^2 - 4 $
# Answer:
$ y = (x + 1)^2 - 4 $

### Problem 4: $ y = x^2 - 20x + 101 $
# Explanation:
## Step 1: Complete the square for $ x^2 - 20x $
Coefficient of $ x $ is -20. Half of -20 is -10, squaring -10 gives 100. Add and subtract 100.
$ y = (x^2 - 20x + 100) - 100 + 101 $
## Step 2: Factor and simplify
$ x^2 - 20x + 100 = (x - 10)^2 $, and -100 + 101 = 1.
$ y = (x - 10)^2 + 1 $
# Answer:
$ y = (x - 10)^2 + 1 $

### Problem 5: $ y = 2x^2 + 8x + 1 $
# Explanation:
## Step 1: Factor out the coefficient of $ x^2 $ from the first two terms
Factor out 2 from $ 2x^2 + 8x $: $ y = 2(x^2 + 4x) + 1 $
## Step 2: Complete the square inside the parentheses
For $ x^2 + 4x $, half of 4 is 2, squaring 2 gives 4. Add and subtract 4 inside the parentheses (but multiply the subtracted 4 by 2 since there's a factor of 2 outside).
$ y = 2[(x^2 + 4x + 4) - 4] + 1 $
## Step 3: Distribute and simplify
$ y = 2(x + 2)^2 - 8 + 1 $
$ y = 2(x + 2)^2 - 7 $
# Answer:
$ y = 2(x + 2)^2 - 7 $

### Problem 6: $ y = -3x^2 + 18x + 5 $
# Explanation:
## Step 1: Factor out the coefficient of $ x^2 $ from the first two terms
Factor out -3 from $ -3x^2 + 18x $: $ y = -3(x^2 - 6x) + 5 $
## Step 2: Complete the square inside the parentheses
For $ x^2 - 6x $, half of -6 is -3, squaring -3 gives 9. Add and subtract 9 inside the parentheses (multiply the subtracted 9 by -3 since there's a factor of -3 outside).
$ y = -3[(x^2 - 6x + 9) - 9] + 5 $
## Step 3: Distribute and simplify
$ y = -3(x - 3)^2 + 27 + 5 $
$ y = -3(x - 3)^2 + 32 $
# Answer:
$ y = -3(x - 3)^2 + 32 $

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QUESTION IMAGE

Question

Explanation:

Problem 1: \( y = x^2 - 12x - 25 \)

Step 1: Complete the square for \( x^2 - 12x \)

Step 2: Factor the perfect square trinomial

Step 1: Complete the square for \( x^2 + 4x \)

Step 2: Factor and simplify

Step 1: Complete the square for \( x^2 + 2x \)

Step 2: Factor and simplify

Answer:

Problem 2: \( y = x^2 + 4x + 5 \)