Ginger was asked to solve 25x7 - 196 = 0 by factoring using the
difference of squares pattern. What is/are the correct solution(s)?
(1 point)
• х =
14
, Х
5
14
5
There is no solution for x because the difference of squares pattern does not apply.
• х
14
5
Х =
196
25
There is no solution for x because the difference of squares pattern does not apply.
To solve the equation 25x7 - 196 = 0 using the difference of squares pattern, we need to rewrite it in the form of a squared term minus another squared term.
25x7 - 196 can be rewritten as (5x)^2 - 14^2.
Now we have a difference of squares, where (5x)^2 is the square of 5x and 14^2 is the square of 14.
So, using the difference of squares pattern, we can factor 25x7 - 196 as (5x + 14)(5x - 14) = 0.
Now, to find the values of x, we set each factor equal to zero:
5x + 14 = 0
5x = -14
x = -14/5
and
5x - 14 = 0
5x = 14
x = 14/5
Therefore, the correct solutions are x = -14/5 and x = 14/5.
To solve the equation 25x7 - 196 = 0 using the difference of squares pattern, we need to first understand what the difference of squares pattern is.
The difference of squares pattern applies when we have an expression in the form a^2 - b^2. It can be factored as (a + b)(a - b).
Now let's apply the difference of squares pattern to the given equation:
25x^2 - 196 = 0
Notice that 25 is a perfect square (5^2) and 196 is also a perfect square (14^2). So we can rewrite the equation as:
(5x)^2 - 14^2 = 0
Using the difference of squares pattern, we can factor it as:
(5x + 14)(5x - 14) = 0
Now we solve for x by setting each factor equal to 0:
5x + 14 = 0 or 5x - 14 = 0
Solving the first equation, 5x + 14 = 0, we subtract 14 from both sides:
5x = -14
Dividing both sides by 5, we get:
x = -14/5 or x = -2.8 (rounded to one decimal place)
Solving the second equation, 5x - 14 = 0, we add 14 to both sides:
5x = 14
Dividing both sides by 5, we get:
x = 14/5 or x = 2.8 (rounded to one decimal place)
So the correct solutions are:
x = -2.8 and x = 2.8