The sum of a positive number and 5 is 806 times the reciprocal of the number. Find the number.
note that x>0
(x) + 5 = 806 * (1/x) is your set up
Simplify:
x + 5 = (806/x)
Multiply both sides by x
x(x+5) = (806/x) * x
x^2 + 5x = 806
Set to zero
(x^2 + 5x - 806) = 0
Factor! :D
(x-26)(x+31) = 0
Set each factor to 0
(x-26) = 0
x = 26
(x+31) = 0
x = -31
so, x = {26,31} BUT!...
Remember that your condition is x>0
So your only answer is 26.
Hope this help :^)
thankyou
Well, if the sum of a positive number and 5 is 806 times the reciprocal of the number, let's call the number "x". So, we have the equation x + 5 = 806(1/x).
Now, let's put our detective skills to work. To solve the equation, we can start by multiplying both sides by x to get rid of the fraction. We'll have x^2 + 5x = 806.
Now, let me tell you a little math joke. Why don't scientists trust atoms?
Because they make up everything!
Alright, back to solving the equation. To continue, we can rearrange it to get a quadratic equation: x^2 + 5x - 806 = 0.
We could use the quadratic formula to solve it, but I have another idea. How about we factorize it? After all, factoring can be a lot of fun!
Let's see... We want to find two numbers that multiply to -806 and add up to 5. Quite a challenging task! After a bit of trial and error, we get (26, -31).
So, the equation factors into (x + 26)(x - 31) = 0. This means that either x + 26 = 0 or x - 31 = 0.
Solving those equations gives us x = -26 or x = 31. But we're looking for a positive number, so the answer is x = 31.
So, the number we're looking for is 31. Ta-da!
To find the number, let's denote it as x.
According to the given information, the sum of the positive number x and 5 is equal to 806 times the reciprocal of the number.
Mathematically, this can be written as:
x + 5 = 806 * (1/x)
To solve this equation, we can start by multiplying both sides by x to get rid of the denominator:
x(x + 5) = 806
Expanding the left side of the equation:
x^2 + 5x = 806
Rearranging the equation to bring all terms to one side:
x^2 + 5x - 806 = 0
Next, we can try to factor the quadratic equation. However, this equation doesn't factor easily, so we'll use the quadratic formula to solve for x:
x = (-b ± √(b^2 - 4ac)) / 2a
For our equation a = 1, b = 5, and c = -806.
Substituting these values into the quadratic formula:
x = (-5 ± √(5^2 - 4 * 1 * -806)) / (2 * 1)
Simplifying:
x = (-5 ± √(25 + 3224)) / 2
x = (-5 ± √3249) / 2
x = (-5 ± 57) / 2
Now, we have two solutions for x:
x1 = (-5 + 57) / 2
x1 = 52 / 2
x1 = 26
x2 = (-5 - 57) / 2
x2 = -62 / 2
x2 = -31
Since a positive number is required, the solution is x = 26.
To find the number, let's first represent the number as "x".
According to the given information, the sum of a positive number (x) and 5 is equal to 806 times the reciprocal of the number. Mathematically, it can be expressed as:
x + 5 = 806 * (1/x)
Now let's solve this equation step by step to find the value of x:
1. Multiply both sides of the equation by x to eliminate the denominator:
x * (x + 5) = 806
2. Expand the left side of the equation:
x^2 + 5x = 806
3. Rewrite the equation in standard quadratic form:
x^2 + 5x - 806 = 0
Now we have a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. Since factoring may be challenging in this case, let's use the quadratic formula:
4. Apply the quadratic formula, which states:
For an equation "ax^2 + bx + c = 0", the solutions for x can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = 1, b = 5, and c = -806. Substituting these values into the quadratic formula:
x = (-5 ± √(5^2 - 4 * 1 * -806)) / (2 * 1)
5. Simplify the expression inside the square root:
x = (-5 ± √(25 + 3224)) / 2
x = (-5 ± √3249) / 2
Notice that √3249 = 57, because 57 * 57 = 3249.
6. Solve for both values of x:
x = (-5 + 57) / 2 = 52 / 2 = 26
x = (-5 - 57) / 2 = -62 / 2 = -31
Given that the question states "a positive number," we can disregard the negative value (-31), leaving us with the final answer:
The positive number is 26.