** this is a 2-page docunne factor the perfect square trinomial, then solve the quadratic equation by square roots. 1. $x^{2}+18x + 81 = 1$ 2. $x^{2}-10x + 25 = 64$ 3. $x^{2}-16x + 64 = 4$ 4. $x^{2}+2x + 1 = 49$ solve each quadratic equation by completing the square. simplify all irrational solutions. 5. $x^{2}-6x - 16 = 0$ 6. $x^{2}-20x + 19 = 0$ 7. $x^{2}+4x + 14 = 46$ 8. $x^{2}+14x + 32 = -8$ © gina wilson (all things algebra®, llc), 2012 - 2017

Question

Accepted Answer

### Problem 1: $ x^2 + 18x + 81 = 1 $
# Explanation:
## Step 1: Factor the perfect square trinomial
The left - hand side $ x^2+18x + 81 $ is a perfect square trinomial. We know that $ (a + b)^2=a^{2}+2ab + b^{2} $. For $ x^2+18x + 81 $, if $ a=x $ and $ 2ab = 18x $, then $ b = 9 $ (since $ 2\times x\times9=18x $) and $ b^{2}=81 $. So $ x^2 + 18x+81=(x + 9)^2 $. The equation becomes $ (x + 9)^2=1 $
## Step 2: Solve by square roots
Take the square root of both sides: $ x + 9=\pm\sqrt{1}=\pm1 $
## Step 3: Solve for $ x $
Case 1: $ x+9 = 1 $, then $ x=1 - 9=-8 $
Case 2: $ x + 9=-1 $, then $ x=-1-9=-10 $
# Answer: $ x=-8 $ or $ x = - 10 $

### Problem 2: $ x^2-10x + 25=64 $
# Explanation:
## Step 1: Factor the perfect square trinomial
The left - hand side $ x^2-10x + 25 $ is a perfect square trinomial. Using the formula $ (a - b)^2=a^{2}-2ab + b^{2} $, if $ a=x $, $ 2ab = 10x $, then $ b = 5 $ (since $ 2\times x\times5 = 10x $) and $ b^{2}=25 $. So $ x^2-10x + 25=(x - 5)^2 $. The equation becomes $ (x - 5)^2=64 $
## Step 2: Solve by square roots
Take the square root of both sides: $ x - 5=\pm\sqrt{64}=\pm8 $
## Step 3: Solve for $ x $
Case 1: $ x-5 = 8 $, then $ x=8 + 5=13 $
Case 2: $ x - 5=-8 $, then $ x=-8 + 5=-3 $
# Answer: $ x = 13 $ or $ x=-3 $

### Problem 3: $ x^2-16x + 64 = 4 $
# Explanation:
## Step 1: Factor the perfect square trinomial
The left - hand side $ x^2-16x + 64 $ is a perfect square trinomial. Using $ (a - b)^2=a^{2}-2ab + b^{2} $, if $ a=x $, $ 2ab=16x $, then $ b = 8 $ (since $ 2\times x\times8=16x $) and $ b^{2}=64 $. So $ x^2-16x + 64=(x - 8)^2 $. The equation becomes $ (x - 8)^2=4 $
## Step 2: Solve by square roots
Take the square root of both sides: $ x - 8=\pm\sqrt{4}=\pm2 $
## Step 3: Solve for $ x $
Case 1: $ x-8 = 2 $, then $ x=2 + 8 = 10 $
Case 2: $ x - 8=-2 $, then $ x=-2+8 = 6 $
# Answer: $ x = 10 $ or $ x=6 $

### Problem 4: $ x^2+2x + 1=49 $
# Explanation:
## Step 1: Factor the perfect square trinomial
The left - hand side $ x^2+2x + 1 $ is a perfect square trinomial. Using $ (a + b)^2=a^{2}+2ab + b^{2} $, if $ a=x $, $ 2ab = 2x $, then $ b = 1 $ (since $ 2\times x\times1=2x $) and $ b^{2}=1 $. So $ x^2+2x + 1=(x + 1)^2 $. The equation becomes $ (x + 1)^2=49 $
## Step 2: Solve by square roots
Take the square root of both sides: $ x + 1=\pm\sqrt{49}=\pm7 $
## Step 3: Solve for $ x $
Case 1: $ x + 1=7 $, then $ x=7-1 = 6 $
Case 2: $ x + 1=-7 $, then $ x=-7 - 1=-8 $
# Answer: $ x = 6 $ or $ x=-8 $

### Problem 5: $ x^2-6x-16 = 0 $ (solving by completing the square)
# Explanation:
## Step 1: Move the constant term to the right
$ x^2-6x=16 $
## Step 2: Complete the square
The coefficient of $ x $ is $ - 6 $. Half of $ - 6 $ is $ - 3 $, and $ (-3)^2 = 9 $. Add 9 to both sides of the equation: $ x^2-6x + 9=16 + 9 $
## Step 3: Factor the left - hand side
The left - hand side $ x^2-6x + 9=(x - 3)^2 $, and the right - hand side is $ 25 $. So $ (x - 3)^2=25 $
## Step 4: Solve by square roots
Take the square root of both sides: $ x - 3=\pm\sqrt{25}=\pm5 $
## Step 5: Solve for $ x $
Case 1: $ x-3 = 5 $, then $ x=5 + 3=8 $
Case 2: $ x - 3=-5 $, then $ x=-5 + 3=-2 $
# Answer: $ x = 8 $ or $ x=-2 $

### Problem 6: $ x^2-20x + 19 = 0 $ (solving by completing the square)
# Explanation:
## Step 1: Move the constant term to the right
$ x^2-20x=-19 $
## Step 2: Complete the square
The coefficient of $ x $ is $ - 20 $. Half of $ - 20 $ is $ - 10 $, and $ (-10)^2 = 100 $. Add 100 to both sides: $ x^2-20x+100=-19 + 100 $
## Step 3: Factor the left - hand side
The left - hand side $ x^2-20x + 100=(x - 10)^2 $, and the right - hand side is $ 81 $. So $ (x - 10)^2=81 $
## Step 4: Solve by square roots
Take the square root of both sides: $ x - 10=\pm\sqrt{81}=\pm9 $
## Step 5: Solve for $ x $
Case 1: $ x-10 = 9 $, then $ x=9 + 10=19 $
Case 2: $ x - 10=-9 $, then $ x=-9 + 10 = 1 $
# Answer: $ x = 19 $ or $ x = 1 $

### Problem 7: $ x^2+4x + 14=46 $ (solving by completing the square)
# Explanation:
## Step 1: Move the constant term to the right
$ x^2+4x=46 - 14 $, so $ x^2+4x=32 $
## Step 2: Complete the square
The coefficient of $ x $ is $ 4 $. Half of $ 4 $ is $ 2 $, and $ 2^2 = 4 $. Add 4 to both sides: $ x^2+4x + 4=32 + 4 $
## Step 3: Factor the left - hand side
The left - hand side $ x^2+4x + 4=(x + 2)^2 $, and the right - hand side is $ 36 $. So $ (x + 2)^2=36 $
## Step 4: Solve by square roots
Take the square root of both sides: $ x + 2=\pm\sqrt{36}=\pm6 $
## Step 5: Solve for $ x $
Case 1: $ x+2 = 6 $, then $ x=6 - 2=4 $
Case 2: $ x + 2=-6 $, then $ x=-6 - 2=-8 $
# Answer: $ x = 4 $ or $ x=-8 $

### Problem 8: $ x^2+14x + 32=-8 $ (solving by completing the square)
# Explanation:
## Step 1: Move the constant term to the right
$ x^2+14x=-8 - 32 $, so $ x^2+14x=-40 $
## Step 2: Complete the square
The coefficient of $ x $ is $ 14 $. Half of $ 14 $ is $ 7 $, and $ 7^2 = 49 $. Add 49 to both sides: $ x^2+14x + 49=-40+49 $
## Step 3: Factor the left - hand side
The left - hand side $ x^2+14x + 49=(x + 7)^2 $, and the right - hand side is $ 9 $. So $ (x + 7)^2=9 $
## Step 4: Solve by square roots
Take the square root of both sides: $ x + 7=\pm\sqrt{9}=\pm3 $
## Step 5: Solve for $ x $
Case 1: $ x+7 = 3 $, then $ x=3 - 7=-4 $
Case 2: $ x + 7=-3 $, then $ x=-3 - 7=-10 $
# Answer: $ x=-4 $ or $ x=-10 $

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 1: \( x^2 + 18x + 81 = 1 \)

Step 1: Factor the perfect square trinomial

Step 2: Solve by square roots

Step 3: Solve for \( x \)

Step 1: Factor the perfect square trinomial

Step 2: Solve by square roots

Step 3: Solve for \( x \)

Step 1: Factor the perfect square trinomial

Step 2: Solve by square roots

Step 3: Solve for \( x \)

Answer:

Problem 2: \( x^2-10x + 25=64 \)