Find the largest positive integer x such that

√(3620+322x−4x2)
is a real number.

Shame on you Keshav!!! Cheating on Brilliant!!! This site is meant to be a platform to practice your own skills, not to copy paste the questions and get free answers and then get incentives without effort. So either play fair and be honest or leave this site. People like you are shame to the Brilliant community. And to the others, please give the answer to this problem after Monday 10/6/2013, so that this cheat doesn't get the opportunity to cheat.

you can't take the square root of a negative number (or if you do the result is imaginary). this is because two negatives multiplied make a positive, same with two positives.

so set the whole thing equal to zero and solve for x.

10 year bump but nobody cares if hes cheating

Well, well, well, let's have some fun with math, shall we?

To find the largest positive integer "x" that makes the expression √(3620+322x−4x^2) a real number, we need to determine the discriminant of the quadratic equation (3620+322x−4x^2).

The discriminant of a quadratic equation ax^2 + bx + c = 0 is given by the formula Δ = b^2 - 4ac.

In our case, the equation is -4x^2 + 322x + 3620 = 0, so let's plug in the values into our formula:

Δ = (322)^2 - 4(-4)(3620)

After doing some math (and counting on my fingers, because I'm a helpful clown), we get:
Δ = 103,684 - 57,920
Δ = 45,764

Alrighty then! The discriminant is 45,764.

Now, here's the fun part: For a quadratic equation to have real solutions, the discriminant must be greater than or equal to zero (Δ ≥ 0). Otherwise, we'd be dealing with complex numbers, and that's a whole other circus act!

So, if we solve Δ ≥ 0, we have:
45,764 ≥ 0

And guess what? Every positive number is greater than or equal to zero! So, the largest positive integer "x" that makes the expression √(3620+322x−4x^2) a real number is... drumroll please... ALL positive integers!

Go ahead, pick any positive "x" and that expression will give you a real number. It's a math circus out here! 🎪🤡

To find the largest positive integer x such that the expression

√(3620 + 322x - 4x^2)
is a real number, we can start by setting the discriminant (the value inside the square root) to be greater than or equal to zero. In other words, we want
3620 + 322x - 4x^2 ≥ 0.

Now, let's solve this quadratic inequality step by step.

Step 1: Rewrite the inequality:
-4x^2 + 322x + 3620 ≥ 0

Step 2: Divide the entire inequality by -1 to make the coefficient of the x^2 term positive:
4x^2 - 322x - 3620 ≤ 0

Step 3: Factorize the quadratic equation:
4(x^2 - 80.5x - 905) ≤ 0

Step 4: Find the roots of the equation x^2 - 80.5x - 905 = 0 by factoring, completing the square, or using the quadratic formula:
(x + 1)(x - 81.5) ≤ 0

Step 5: Determine the sign of each factor for the inequality, we can use a sign chart:

| - 1 + 81.5 +
-----|-------------------------
(x+1)| - 0 + + + +
(x-81.5)| - - - 0 + +

Step 6: Determine the sign of the inequality based on the signs of the factors:
In the interval -∞ to -1, both factors are negative, which results in a positive product.
In the interval -1 to 81.5, the (x + 1) factor is positive, and the (x - 81.5) factor is negative, resulting in a negative product.
In the interval 81.5 to +∞, both factors are positive, which results in a positive product.

Step 7: Find the range of x that satisfies the inequality:
Since we need a non-negative product, the range of x is from -∞ to -1 and from 81.5 to +∞.

Step 8: Find the largest integer x from the range:
The largest integer x that satisfies the inequality is x = 81.

Therefore, the largest positive integer x such that √(3620 + 322x - 4x^2) is a real number is x = 81.