solve x+x^.5=23?

x + x^0.5 = 23

x + √x = 23

√x = 23 - x, where 0 ≤ x ≤ 23

x = (23 - x)^2

x = 529 - 46x + x^2

x^2 - 47x + 529 = 0

x = (1/2)(47 ± √93)

but since x ≥ 23,

x = (1/2)(47 - √93)

Edit: more details for finding the domain:

We have √x = 23 - x,

the expression under the square root sign must be equal to or greater than zero => x ≥ 0

and the value of a square root of a value is always greater than or equal to zero => 23 - x ≥ 0

x ≥ 0 and 23 - x ≥ 0 gives us the domain : 0 ≤ x ≤ 23

x = (1/2)(47 ± √93)

x ≈ 16.67, 28.32 , now you see that only x = 16.67 is the domain.

=> x = (1/2)(47 - √93)

I'm afraid the Quadratic Formula will be as much use as a lead balloon with the equation in that form !. Rearrange that equation as x6(1/2) = 23 - x and since x^(1/2) means the square root of x, we can square both sides of that modified equation to get x = (23 - x)^2 or x = 529 - 46x + x^2. Now subtract x from both sides and we get x^2 - 47x + 529 = 0, so now we can use the Quadratc Formula and we will get x = [47 +/- sqrt((-47)^2 - 4*529)]/2 and thatreduces to x = [47 +/- sqrt 93]/2 which gives you two solutions.

There is an easier way.

Let y^2 = x

y^2 + (y^2)^1/2 = 23 Subtract 23

y^2 + y - 23 = 0 Now solve for y

a = 1

b = 1

c =-23

y = [-b +/- sqrt(b^2 - 4ac)]/(2a)

y = [-1 +/- sqrt(1 - (-92) ) ]/2

y = [-1 +/- sqrt(93)]/2

y = -0.5 +/- 0.5 sqrt(93)

x = y^2 so you have to square the result above.

x = (-0.5 +/- 0.5sqrt(93))^2

x = 0.25 +/- 2*(-0.5)(0.5) (sqrt(93) + 0.25*93

x = 0.25 -/+ (0.5)-sqrt(93) + 23.25

x = 23.50 -/+ (-0.5)sqrt(93)

x = 23.5 - 0.5*sqrt(93)

x1 = 23.5 + 0.5* sqrt(93)

With such nasty numbers, you need to check this.

x + sqrt(x) = 23

23.5 - 0.5*sqrt(93) + sqrt(23.5 - 0.5*sqrt(93)) = 23

I'm just going to put in numbers to see what happens.

23.5 - 4.82 + sqrt(23.5 - 4.82))= ? 23

23.5 - 4.82 + sqrt(18.68) =? 23

18.68 + 4.32 =? 23 which checks

23 = 23 check

x + sqrt(x) = 23

23.5 + 0.5* sqrt(93) + sqrt(23.5 + 0.5* sqrt(93) ) There is no way that this will come back to 23. The numbers behind 23.5 are all plus and 23.5 is greater than 23 already so this solution is not a solution.

x^1/2 +x -23 = 0

Use quadratic formula now.

http://www.mathsisfun.com/algebra/quadratic-equati...