given that a and b are rational numbers, find the value of a and of b in the following equation:
a-bsquareroot7 = (2+squareroot7)(2-squareroot7)+squareroot7
a-broot(7)=(2)^2-root(7)^2+roo(7)
a-broot(7)=4-7+root(7)
a-broot(7)=root(7)-3
now just compare the coefficent
a=?
b=?
(2+√7)(2-√7)+√7
4-7+√7
-3+√7
a = -3
b = -1
To find the values of a and b in the equation, we can simplify the right-hand side of the equation first.
Let's expand the right-hand side using the difference of squares formula: (a - b)(a + b) = a^2 - b^2.
So, (2 + sqrt(7))(2 - sqrt(7)) = 2^2 - (sqrt(7))^2 = 4 - 7 = -3.
Now our equation becomes:
a - b√7 = -3 + √7
To find the values of a and b, we need to separate the terms with √7 from the terms without √7.
We can rewrite the equation as:
a = -3 + √7 + b√7
Now, we have the following system of equations:
a = -3 + √7 + b√7
0 = b
Since a and b are rational numbers, we know that b must be 0 (as there is no rational multiple of √7 that can be added to -3 + √7 to result in a rational number). Therefore, b = 0.
Substituting b = 0 back into our original equation, we get:
a - 0√7 = -3 + √7
a = -3 + √7
So, the value of b is 0, and the value of a is -3 + √7.