Out of group of goats,Two times the square root of the number of goats were kept in shade, 1/4 of the number were taken for grazing and 15 were sold, find the initial no. Of goats?
N=2sqrtN +.25N+15
change variables to ease tension
x=sqrtN
.75x^2 -2x-15=0
Ahhhhah! a quadratic equation.
x= (2+-sqrt(4+4*3/4*15))/1.5
x=(2+-sqrt(49))/1.5
x=9/1.5=6
x^2=N=36
Thanks
To solve this problem, we can break it down into smaller steps.
Step 1: Let's assume the initial number of goats is 'x'.
Step 2: "Two times the square root of the number of goats were kept in shade." This means 2 * √x goats were kept in shade.
Step 3: "1/4 of the number were taken for grazing." This means 1/4 * x goats were taken for grazing.
Step 4: "15 were sold." So, 15 goats were sold.
Step 5: Now we can write an equation based on the given information. The equation will be:
x = 2 * √x + 1/4 * x + 15
Step 6: Simplifying the equation, we can start by multiplying 1/4 * x:
x = 2 * √x + (1/4 * x) + 15
x = 2 * √x + x/4 + 15
Step 7: To eliminate the square root, we can square both sides of the equation:
x^2 = (2 * √x + x/4 + 15)^2
Step 8: Expanding the squared term on the right side:
x^2 = (2 * √x)^2 + (x/4)^2 + 15^2 + 2 * 2 * √x * x/4 + 2 * 2 * √x * 15 + 2 * 15 * x/4
x^2 = 4x + (x^2/16) + 225 + (√x * x/2) + 60√x + 15x/2
Step 9: Simplifying further:
x^2 - 4x - (x^2/16) - 15x/2 = 225 + 60√x - (√x * x/2)
x^2 - (64/16)x - (x^2/16) - 15x/2 + (√x * x/2) + 60√x - 225 = 0
(16x^2 - 64x - x^2 - 120x + 8√x * x + 960√x - 14400) / 16 = 0
Simplifying the equation:
15x^2 - 184x + 8√x * x + 960√x - 14400 = 0
Step 10: At this stage, we can solve the quadratic equation to find the value of x.
The equation is now in the form ax^2 + bx + c = 0, where:
a = 15
b = -184
c = 8√x * x + 960√x - 14400
We can use the quadratic formula to solve for x:
x = (-b ± √(b^2 - 4ac)) / (2a)
Substituting the values:
x = (-(-184) ± √((-184)^2 - 4 * 15 * (8√x * x + 960√x - 14400))) / (2 * 15)
Simplifying:
x = (184 ± √(33856 - 480√x * x - 57600√x + 864000)) / 30
Now, to find the exact value of x, we need to substitute different values for √x and then solve the resulting equation. However, without knowing the value of √x, we cannot find the exact initial number of goats.
Therefore, the initial number of goats cannot be determined without more information.