+23251 If you rewrte x2 = bx - 1 as x2 - bx + 1, then the question can be rewritten as:
For what values of b will the function f(x) = x2 - bx + 1 have: two roots? one root? no roots?
x2 - bx + 1 will be a perfect square if b = -2 or b = 2, giving either f(x) = x2 + 2x + 1 or f(x) = x2 + 2x + 1
When the function is a perfect square, the equation has only one root. One root: b = -2 or 2.
When b is a value between -2 and 2, the equation has no roots. No roots if b is in the interval (-2, 2).
When be is a value that is either larger than 2 or smaller than -2, the equation has no roots. Two roots if b is either in the interval (-infinity, -2) or (2, infinity).
The "no roots", "two roots" answer can be checked by graphing or proven by completing the square.
+23251 If you rewrte x2 = bx - 1 as x2 - bx + 1, then the question can be rewritten as:
For what values of b will the function f(x) = x2 - bx + 1 have: two roots? one root? no roots?
x2 - bx + 1 will be a perfect square if b = -2 or b = 2, giving either f(x) = x2 + 2x + 1 or f(x) = x2 + 2x + 1
When the function is a perfect square, the equation has only one root. One root: b = -2 or 2.
When b is a value between -2 and 2, the equation has no roots. No roots if b is in the interval (-2, 2).
When be is a value that is either larger than 2 or smaller than -2, the equation has no roots. Two roots if b is either in the interval (-infinity, -2) or (2, infinity).
The "no roots", "two roots" answer can be checked by graphing or proven by completing the square.
