Answer :
Final answer:
To solve the problem, we form a system of equations using the given information and then solve it to find the numbers.
Explanation:
Let's represent the two numbers as x and y. We are given that the sum of the numbers is eleven, so we have the equation x + y = 11.
We are also given that the square of one number is 1 more than four times the other number, which can be written as x^2 = 4y + 1.
Now we have a system of equations: x + y = 11 and x^2 = 4y + 1. Solving this system, we can find the values of x and y.
Let's substitute y = 11 - x into the second equation. We have x^2 = 4(11 - x) + 1.
Expanding and rearranging, we get x^2 - 4x - 43 = 0. We can solve this quadratic equation to find the values of x.
Using the quadratic formula, we find that the solutions for x are x = 7 and x = -3. Plugging these values back into the equation x + y = 11, we can find the corresponding values of y. So the numbers are either 7 and 4, or -3 and 14.
Learn more about Solving a system of equations here:
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