Absolute Value Question: Solving |2x - 1| = x^2 with Step-by-Step Guide

[majormuss]

Homework Statement

how do I solve this? I am confused...
|2x - 1| = x^2

Homework Equations

The Attempt at a Solution

when i tried it I ended up with a solution set of (1,-1). But the official answer is quite different so I am confused!

 

[Mentallic]

The absolute value function is defined as follows:

$$|x| = x, x\geq 0$$

$$|x|= -x, x< 0$$

But you have 2x-1 so replace the x in the absolute value for 2x-1. Try this and then solve both equations $2x-1=x^2$ and $-(2x-1)=x^2$. Once you find your solutions, remember to be sure to scrap the solutions that aren't valid. That is, if you get an answer of -10 for $2x-1=x^2$ then you know it's not valid because we are assuming 2x-1>0, or, x>1/2.

 

[Lunat1c]

You could also approach this graphically.

Imagine the line y=2x-1, that's a line with a positive slope and intercepts the y-axis at (0,-1). Now, |2x-1| refers to the absolute values of y (i.e. |y|) so you have to reflect what is below the x-axis in the x-axis itself to get your |y|=|2x-1| which looks like a 'V' shaped graph. The function of the graph that you obtain can be defined as follows:

$$y = 2x-1 for x >= \frac{1}{2}$$

and
$$y = 1 - 2x for x <= \frac{1}{2}$$

Now all you have to do is sketch the graph of $$x^2$$ on top of that and simply determine where your V-shaped graph (y=|2x-1|) is equal to the graph of y=x^2 by finding the points of intersection.