The sum of an integer and 3 times it's square root is 40. What is the integer?

Answer 1

Let's name x the integer.
40=x+3√x
Let's isolate the square root to avoid annoying calculations with the root as coefficient:
40-x=3√x
Let's square both terms, to get rid of the root:
sqr(40-x)=9x
Then:
1600+x*x-80x=9x
A more usable form:
x*x-89x+1600=0
Solving by mean of the quadratic formula:
x = (-b+-√(b*b-4ac))/2a
we get:
x = (89+-√(7921-6400))/2
x = (89+-√1521)/2
x = (89+-39)/2
The roots are:
x1=64
x2=25
To satisfy the total sum of 40, 25 is the only allowable root.
Bye

Answer 2

Let the integer we are looking for be x^2. Then x is the square root of the integer. This simple trick lets us reformulate the question as a simple quadratic equation:
x^2 + 3x = 40, whence:
x^2+3x-40 = 0. This may be factorized as:
(x+8)(x-5) = 0, giving us two solutions
x = -8 and x =5. Thus, the integers we are looking for are either 64 or 25, both of which satisfy your conditions. (Note that sqrt(x) = +/- x.)

Answer 3

let your integer be "x"

therfore the sum (addition) of an integer (x) and 3 times its square root (3*sqrt x) is (equals) 40 ...

becomes:

x + (3) (sqrt x) = 40

3 (sqrt x) = 40 - x
sqrt x = (40 - x) / 3 ... divide both sides by 3
x = (1600 - 80x + x^2) / 9 ... square both sides
9x = x^2 - 80x + 1600 ... multiply both sides by 9
x^2 - 89x + 1600 = 0 ... subtract 9x from both sides

use the quadratic formula to find the solutions to "x" ...

you get:

x = 64
or
x = 25

now check your solutions...

x= 64 :

64 + 3(sqrt 64) = 40
64 + 3(8) = 40
64 + 24 = 40
88 = 40
thefore 64 is NOT a solution for your problem...

now check 25...

x = 25 :

25 + 3(sqrt 25) = 40
25 + 3(5) = 40
25 + 15 = 40
40 = 40
therefore your solution is 40

the hard part might be in setting up the problem, the rest is algebra and using the quadratic formula (unless you can factor it out by hand).

good luck!

Answer 4

Let x = the integer
Therefore, x + 3*sqrt(x) = 40

Subtract x from both sides :
3*sqrt(x) = 40 - x

Square both sides :
9x = 1600 - 80x + x^2

Subtract 9x from both sides and rearrange :
x^2 - 89x + 1600 = 0

Factorise :
(x - 25)(x - 64) = 0

Solutions are : x = 25 or 64

But if x = 64 then 64 + 3*sqrt(64) = 88,
which is not equal to 40.

Thus, the only solution is x = 25 giving :
25 + 3*sqrt(25) = 40.

Answer 5

x+3√x =40

Let z=√x:

z²+3z=40
z²+3z-40=0

Using the quadratic formula:

z=(-3±√(9+160))/2 = (-3±13)/2

However, as square roots are always positive, we may discard the negative solution, so in fact z=10/2=5. Therefore x=25.

Answer 6

so x+3*x^0.5 =40

3x^0.5 = 40-x square this

9x = 1600-80x +x^2

x^2-89x +1600 =0

you find 25

it works 25+3*5 =40