Here, if we write `u=sqrt(x)` we have:

`4x+3sqrtx=1`

`4u^2+3u-1=1`

`(4u-1)(u+1)=0`

So `u = 1/4`; or `u= -1`.

DANGER! Always think carefully about your answer. You can often get answers which are not true solutions.

`sqrt(x)=1/4` means `x= 1/16`

Check by substitution: `4(1/16)+3(1/4)=1`. OK.

But `sqrt(x)=-1` is not possible (`sqrt(x)` is always `≥ 0`).

We conclude there is only one root: `x=1/16`

To give a better idea what our solution looks like, let's have a look at the graph of `y = 4x + 3sqrt(x) - 1`.

The intersection with the x-axis will tell us the solution for the original equation.

0.20.40.60.81-0.21234567-1-2xyOpen image in a new page

Graph of `y = 4x + 3sqrt(x) - 1`

This is an interesting curve since it starts at `(0,-1)` (we cannot have negative `x`-values and the curve does not continue down the `y`-axis). There is one intersection with the x-axis, at `x = 1/16 = 0.0625`.