To determine the value of a² - b², we can use the identity:
a² - b² = (a + b) * (a - b)
Given that a + b equals 5 and ab equals 6, we can calculate the value of a - b:
To do this, we find a by substituting the value of b in the equation a + b = 5:
a = 5 - b
Next, we use the expression for ab and replace a with (5 - b):
(5 - b) * b = 6
Expanding and rearranging the equation, we get:
5b - b² = 6
Now, we apply the identity a² - b² = (a + b) * (a - b) and replace the values:
a² - b² = (a + b) * (a - b) = 5 * (5b - b²)
Now, we need to find the values of b that satisfy the equation 5b - b² = 6:
By factoring the quadratic equation, we get:
(b - 3)(b - 2) = 0
Setting each factor to zero and solving for b, we find two possible values for b: b = 3 and b = 2.
After obtaining the corresponding values of a:
If b = 3, then a = 5 - b = 5 - 3 = 2
If b = 2, then a = 5 - b = 5 - 2 = 3
Now, we can calculate a² - b² for each case:
When b = 3: a² - b² = 2² - 3² = 4 - 9 = -5
When b = 2: a² - b² = 3² - 2² = 9 - 4 = 5
Thus, the potential values of a² - b² are -5 and 5.