'a gardener plants an orchid with 5776 trees. in each row there were as many trees as the number of rows. find the number of rows'

What is the square root of 5,776?

To find the number of rows in the orchid, we can use a quadratic equation.

Let's assume the number of rows as 'n'.

In the first row, there will be 1 tree.
In the second row, there will be 2 trees.
In the third row, there will be 3 trees.
...
In the nth row, there will be n trees.

Now, we know that the total number of trees planted in the orchid is 5776.

So, the equation becomes: 1 + 2 + 3 + ... + n = 5776.

The sum of the first 'n' natural numbers is given by the formula: (n * (n + 1)) / 2.

Using this formula, we can rewrite the equation as: (n * (n + 1)) / 2 = 5776.

To solve this equation, we can multiply both sides by 2 to get rid of the fraction: n * (n + 1) = 2 * 5776.
Simplifying further, we get: n^2 + n = 11552.

Rearranging, we have: n^2 + n - 11552 = 0.

Now, we have a quadratic equation in the form of ax^2 + bx + c = 0, where a = 1, b = 1, and c = -11552.

To solve this equation, we can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a.

Applying the values, we get: n = (-1 ± √(1^2 - 4 * 1 * -11552)) / 2 * 1.

Simplifying further, we get: n = (-1 ± √(1 + 46208)) / 2.

n = (-1 ± √(46209)) / 2.

n = (-1 ± 215) / 2.

Now, we have two possible solutions for 'n':
1) n = (-1 + 215) / 2 = 214 / 2 = 107.
2) n = (-1 - 215) / 2 = -216 / 2 = -108.

Since the number of rows cannot be negative, we can ignore the second solution.

Therefore, the number of rows in the orchid is 107.

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