If a and b are integers, find values such that a^2 - b^2 = 21.
how would you calculate this?
a^2-b^2 = (a-b)(a+b)
since 21 = 3*7,
a-b = 3
a+b = 7
so,
a = 5, b = 2
thank you!!
To find values of a and b that satisfy the equation a^2 - b^2 = 21, you can use the difference of squares formula.
The difference of squares formula states that a^2 - b^2 can be factored as (a + b)(a - b). So, we can rewrite the equation as (a + b)(a - b) = 21.
Now, we need to find integer values for a and b that make this equation true. To do this, we can start by listing all the factor pairs of 21: (1, 21), (-1, -21), (3, 7), and (-3, -7).
For each factor pair, we can set up two equations by letting (a + b) equal one factor and (a - b) equal the other factor. For example:
Equation 1: a + b = 1
Equation 2: a - b = 21
Solving this system of equations, we can add Equation 1 and Equation 2 to eliminate b:
(a + b) + (a - b) = 1 + 21
2a = 22
a = 11
Substituting the value of a into Equation 1, we can solve for b:
11 + b = 1
b = -10
Therefore, one set of values that satisfy the equation a^2 - b^2 = 21 is a = 11 and b = -10.
You can repeat this process for the other factor pairs of 21 to find additional solutions.
To find values of a and b such that a^2 - b^2 = 21, we can use the factoring property of the difference of squares.
The difference of squares formula states that (a^2 - b^2) can be factored as (a + b)(a - b).
So, we can rewrite the equation as (a + b)(a - b) = 21.
Now, we need to find two factors of 21 such that their difference is an even number. This is because if the difference of the two factors is odd, then there is no way to express it as a difference of squares.
The factors of 21 are 1, 3, 7, and 21. We can try different combinations of these factors to find the desired result:
1. (a + b) = 21, (a - b) = 1
Solving these equations simultaneously, we get a = 11, b = 10.
2. (a + b) = 7, (a - b) = 3
Solving these equations simultaneously, we get a = 5, b = 1.
So the possible values of a and b such that a^2 - b^2 = 21 are a = 11, b = 10 or a = 5, b = 1.
You can also verify these solutions by substituting the values of a and b back into the original equation and checking if it holds true.