Final answer:
The sum of two numbers is 6 and the sum of their squares is 28. To find the values of these numbers, we can set up a system of equations and solve it using the quadratic formula. The exact values of the two numbers are (9 - √47) / 2 and (3 + √47) / 2.
Step-by-step explanation:
To find the values of the two numbers, we can set up a system of equations. Let's call the numbers x and y. We are given that the sum of the numbers is 6, so we can write the equation x + y = 6. We are also given that the sum of their squares is 28, so we can write the equation x^2 + y^2 = 28.
We can solve this system of equations by substitution or elimination. Let's solve it by substitution. We can rearrange the first equation to solve for one variable, like x = 6 - y. Substituting this expression for x into the second equation, we get (6 - y)^2 + y^2 = 28. Expanding and simplifying, we get y^2 - 6y + 9 + y^2 = 28. Combining like terms, we have 2y^2 - 6y + 9 = 28. Rearranging, we get 2y^2 - 6y - 19 = 0.
This is a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. In this case, the quadratic equation does not factor easily, so we'll use the quadratic formula. The quadratic formula is given by x = (-b ± √(b^2 - 4ac)) / (2a), where a, b, and c are the coefficients of the quadratic equation. In our case, a = 2, b = -6, and c = -19. Plugging these values into the quadratic formula, we get y = (6 ± √(6^2 - 4(2)(-19))) / (2(2)). Simplifying further, we have y = (6 ± √(36 + 152)) / 4, which simplifies to y = (6 ± √188) / 4. We can further simplify this expression by dividing the numerator and denominator by 4, giving us y = (3 ± √47) / 2. So the two possible values of y are (3 + √47) / 2 and (3 - √47) / 2.
Now, we can substitute these values of y into the first equation x + y = 6 to find the corresponding values of x. If y = (3 + √47) / 2, we have x + (3 + √47) / 2 = 6. Multiplying both sides by 2 to clear the fraction, we get 2x + 3 + √47 = 12. Subtracting 3 and √47 from both sides, we have 2x = 12 - 3 - √47, which simplifies to 2x = 9 - √47. Dividing both sides by 2, we get x = (9 - √47) / 2. Similarly, if y = (3 - √47) / 2, we have x + (3 - √47) / 2 = 6. Multiplying both sides by 2, we get 2x + 3 - √47 = 12. Subtracting 3 and √47 from both sides, we have 2x = 12 - 3 + √47, which simplifies to 2x = 9 + √47. Dividing both sides by 2, we get x = (9 + √47) / 2.
So the exact values of the two numbers are:
x = (9 - √47) / 2 and y = (3 + √47) / 2