Perfect Square Question for Algebra 1

Question

1. What could be added to the expression to make it a perfect square? x^2 + 12x
2. What could be added to the expression to make this example a perfect square? x^2 - 6x + 1
3. The solutions to this equation are 3/4 and 1/4. Explain how to find those solutions by completing the square. x^2 - x + 3/16 = 0
4. Find the value of c to make the expression a perfect square. 4x^2 - 28x + c
5. Using the answer you found for the problem above, write an equivalent expression in the form of squares factors. (kx + m)^2

Answer 1

Hello Cameron, here's my perspective.

A perfect square has the form (kx + b)², which expands to k²x² + 2kbx + b². So, for question 1, we must add something to x² + 12x so the result has the form k²x² + 2kbx + b². Let's assume we want to add just a constant c and no extra "x" term. Then, we want c such that x² + 12x + c has the form k²x² + 2kbx + b². Matching the squared terms gives k=1. Then, matching the x terms while using k=1 gives b=6. Last, matching the constant terms using b=6 gives c=36. So, we must add 36 to x + 12x to make it a perfect square.....the perfect square (x + 6)².

You can solve #2, 4, and 5 using this same reasoning. Here are the answers you'll get:

#2 Add 8

#4 c = 49

#5 (2x - 7)²

As for #3, you're asked to solve x² - x + 3/16 = 0 by completing the square. This means you must first find a number c such that x² - x + c is a perfect square. Based on the above discussion, we know that c=1/4, and the perfect square is (x - 1/2)². Next, use this knowledge to solve x² - x + 3/16 = 0 by completing the square as follows:

x² - x + 1/4 - 1/4 + 3/16 = 0. (Add and subtract c)

(x - 1/2)² - 1/4 + 3/16 = 0. (Express x² - x + 1/4 as a perfect square)

(x - 1/2)² - 1/16 = 0. (Simplify the numbers)

(x - 1/2)² = 1/16 (Move the number to the right side)

(x - 1/2) = 1/4 or -1/4 (Square root both sides)

x = 1/2 + 1/4 or x = 1/2 - 1/4 (Solve for x)

x = 3/4 or x = 1/4

I hope that helped!

Answer 2

Cameron:

Notice (x+6)² = x^2+12x+36

(x-3)² = x^2 -6x + 9 try( x^2-6x+1)+8

try the other problems and let me know if you are stuck