Real Roots

the equation in the square root must positive, so the equation 3620 + 322x -4x^{2} must equal or larger than zero. so we solve equation 3620 + 322x -4x^{2} = 0 , the solution is x1= -10, x2=90.5 . so to the equation positive x must be in the range [-10,90.5]. Thus the largest positive integer x is 90.

$\sqrt{3620 + 322x - 4x^{2}}$ will be a real number when $\boxed{3620 + 322x - 4x^{2} >= 0 }$

Therefore, solve equation $\boxed{3620 + 322x - 4x^{2} = 0 }$ , and you get x = 90.5. All integers less than 90.5 gives real numbers and integers greater than 90.5 give complex numbers.

So, largest positive interger is 90.

We want $x$ to be positive, but we want it to be the largest. Finding x-intercepts will give us the largest that will give us a positive number or 0. So set everything inside the square root equal to 0.

$3620+322x-4x^{2}=0$

$-2(x+10)(2x-181)=0$

$x=-10, 90.5$

$x$ has to be positive, so -10 is out of the picture. But 90.5 is not an integer! So we will have to find the integer closest to but smaller than 90.5, which is $\boxed{90}$

First, I thought I'd solve the quadratic under the square root for zero. I worked out that this was 90.5. A square root of 0 does not result in a real number as a solution. Neither does a square root of a negative . Therefore $3620 + 322x - 4x^{2}$ must be a postive . From this I worked out the closest integer to 90.5 that would result in a postive value, was 90.

The roots of the equation are -10 and 90.5.

But these values evaluate the equation to zero.

90 is the largest possible integer which will satisfy the conditions given .

Hence the Answer.!!!

as coefficient of x^2 is negative, the graph is a downward parabola, hence max value of x would be the greatest root of equation 3620+322x-4x^2 = 0, which is 181/2, and the nearest integer in 90

find the roots of equation(3620+322x-4x^2). x1 will be 90.5 and x2 will be -10. now put the value of x=90.5 and x=-10. the value of whole function at x=-10 is not real. and again we put x=90.5.which gives real value of function and greatest integer of 90.5 is either 90 or 91. now put 90. hence it is real at 90.

For root(3620+322x-4x|2|)to be real the expression must be >=0.Part 322x-4x|2| {= 2x(161-2x)} implies that the greater the value of x the lesser the value of the expression will be. But the least value of the expression is 0. So the least value for x can be got from the equation : 3620+322x-4x|2|=0. This simplifies to x=90.5(greater root). But we can take only integers. So the answer is [x] = 90.

Start by solving the quadratic eq. under the root sign, with this you will obtain the two roots or we can say values of 'x' i.e x=-10 & x=90.5. As the ques says 'x' needs to be a positive integer the ans is 90. You can also verify it by substituting 90 in the place of 'x', you will get a real no. but if you substitute 91 you will get an imaginary no.