The factored form of the expression x2 + (3/7)x + (9/196) is:
x² + (3/7)x + (9/196) = (x + 3/14)(x + 3/196)
Using the factoring method:
The expression can be factored as follows:
x² + (3/7)x + (9/196) = (x + a)(x + b)
To find the values of a and b, we need to find two numbers whose sum is equal to (3/7) and whose product is equal to (9/196).
find the values of a and b:
a + b = 3/7
ab = 9/196
To simplify the process, multiply both sides of the first equation by 7:
7a + 7b = 3
Solve the system of equations:
ab = 9/196
7a + 7b = 3
From the first equation, we can express b in terms of a:
b = (9/196) / a
Substituting this value of b into the second equation:
7a + 7((9/196) / a) = 3
Multiplying both sides by a:
7a² + 7(9/196) = 3a
Multiplying both sides by 196:
1372a² + 63 = 588a
Rearranging the equation:
1372a2 - 588a + 63 = 0
Now, we can solve this quadratic equation to find the values of a and b.
Using the quadratic formula:
a = (-b ± √(b² - 4ac)) / (2a)
a = (-(-588) ± √((-588)² - 4 × 1372 × 63)) / (2 × 1372)
Simplifying:
a = (588 ± √(345744 - 345744)) / 2744
a = (588 ± √0) / 2744
Since the discriminant is zero, there is only one solution for a:
a = 588 / 2744
Simplifying:
a = 3/14
Now, substituting this value of a back into the equation b = (9/196) / a:
b = (9/196) / (3/14)
b = (9/196) × (14/3)
b = 3/196
Therefore, the factored form of the expression x² + (3/7)x + (9/196) is:
x² + (3/7)x + (9/196) = (x + 3/14)(x + 3/196)