Question

Graph y = √x + 2 and y = 3x on the same set of coordinate axes. Use the graphs to find the exact solutions to the equation √x + 2 = 3x. Then, find the solutions using an algebraic method.

Answer 1

x = 1

Explanation:

The graph is shown below. The solution is x = 1.

___

√x +2 = 3x . . . . given

√x = 3x -2 . . . . . subtract 2

x = 9x^2 -12x +4 . . . . square both sides

9x^2 -13x +4 = 0 . . . . subtract x to put in standard form

(x -1)(9x -4) = 0 . . . . . factor the quadratic

x = 1 . . . or . . . x = 4/9 . . . . values of x that make the factors zero

The "solution" x = 4/9 is extraneous, in that it does not satisfy the equation. It would correspond to the intersection of the line with the negative branch of the square root function. That is, it does satisfy -√x +2 = 3x.

The solution is x=1.