Derive the square root of 622521
To find the square root of 622521, we can use the long division method.
Let's start by assuming the square root to be a two-digit number that starts with a digit smaller than or equal to 9. Let's assume the square root is 80.
Taking the square of 80, we get 6400. Since 6400 is smaller than 622521, we need to find the largest digit to append to 80 (our assumption of the square root) to make the new number less than or equal to 622521.
Let's assume the largest digit is x.
So, the equation becomes (80+ x)•(80+ x) ≤ 622521.
Expanding the equation: 6400 + 80x + 80x + x² ≤ 622521.
Simplifying: 2x² + 160x + 6400 ≤ 622521.
Rearranging: 2x² + 160x - 616121 ≤ 0.
Now, we can solve this quadratic equation.
Using the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a), where a = 2, b = 160, c = -616121.
x = (-160 ± √((160)² - 4(2)(-616121))) / (2*2).
x = (-160 ± √(25600 + 4928488)) / 4.
x = (-160 ± √(4954088)) / 4.
x = (-160 ± 703.186) / 4.
Now, we have two possible values of x: (-160 + 703.186) / 4 and (-160 - 703.186) / 4.
Calculating both values:
x₁ = (-160 + 703.186) / 4 ≈ 135.796.
x₂ = (-160 - 703.186) / 4 ≈ -215.046.
Since we are looking for a digit smaller than or equal to 9, the value of x will be 0 (since it is the closest value).
Therefore, the square root of 622521 is around 80.