Let's solve problem one part one. We're given that x plus y equals four and x times y equals two. We need to find the value of x minus y squared. We'll use the algebraic identity: x minus y squared equals x plus y squared minus four x y. Substituting our given values: x minus y squared equals four squared minus four times two, which equals sixteen minus eight, giving us eight.
Now let's solve problem one part two. We're given that a plus b squared equals seven and a minus b squared equals three. We need to find a squared plus b squared and a b. First, we add the two equations: a plus b squared plus a minus b squared equals seven plus three, which equals ten. Using the identity that this sum equals two times a squared plus b squared, we get a squared plus b squared equals five. Next, we subtract the second equation from the first: a plus b squared minus a minus b squared equals seven minus three, which equals four. This difference equals four a b, so a b equals one.
Let's solve problem one part three. We have the expression a squared plus b squared minus four a plus two b plus seven, and we need to determine its nature. We'll complete the square for both variables. First, we group the terms: a squared minus four a, plus b squared plus two b, plus seven. Completing the square for the first group gives us a minus two squared minus four. For the second group, we get b plus one squared minus one. Combining everything: a minus two squared plus b plus one squared plus two. Since both squared terms are always non-negative, their sum plus two is always at least two. Therefore, the answer is D: the value is not less than two.
Now let's solve problem two part one. We're given that x squared minus three x plus one equals zero, and we need to find x squared plus one over x squared, and x to the fourth plus one over x to the fourth. First, we divide the equation by x to get x minus three plus one over x equals zero, which gives us x plus one over x equals three. Next, we square both sides: x plus one over x squared equals nine. Expanding this gives x squared plus two plus one over x squared equals nine, so x squared plus one over x squared equals seven. Finally, we square this result: x squared plus one over x squared squared equals forty-nine. This expands to x to the fourth plus two plus one over x to the fourth equals forty-nine, giving us x to the fourth plus one over x to the fourth equals forty-seven.
To summarize what we've learned: We used algebraic identities like the difference of squares to solve for unknown expressions. We applied the completing the square technique to analyze the nature of quadratic expressions. We transformed complex equations using clever substitution methods. These fundamental algebraic techniques form the foundation for more advanced topics in mathematics.