Take any two real numbers whose sum is 1(fractions, decimals, integers, etc. are appropriate. Square the larger and add the smaller. Then square the smaller and the larger.
What will be true?
Let's assume we have two real numbers, a and b, such that a + b = 1.
Step 1: Square the larger number and add the smaller number:
If a > b, then by squaring a, we get a^2, and by adding b, we have a^2 + b.
Step 2: Square the smaller number and add the larger number:
If b > a, then by squaring b, we get b^2, and by adding a, we have b^2 + a.
What will be true?
In general, without any specific information about the values of a and b, it is not possible to determine whether a^2 + b or b^2 + a will be greater. The result will depend on the actual values of a and b.
For example, if a = 0.5 and b = 0.5, then both a^2 + b and b^2 + a will be equal to 0.75.
If a = 0.2 and b = 0.8, then a^2 + b = 0.24 and b^2 + a = 0.84.
If a = -1 and b = 2, then a^2 + b = 3 and b^2 + a = 5.
So, without more specific information, we cannot determine a general rule for which expression will be greater.
To find out what will be true when you square the larger number and add the smaller number, then square the smaller number and add the larger number, let's go step by step:
1. Let's assume the two real numbers whose sum is 1 are represented by a and b, where a is the larger number and b is the smaller number. So we have: a + b = 1.
2. Now, we need to square the larger number (a) and add the smaller number (b): a^2 + b.
3. Next, we square the smaller number (b) and add the larger number (a): b^2 + a.
Now, let's compare these two expressions:
- a^2 + b
- b^2 + a
To determine what will be true, we need to compare these two expressions in different scenarios.
Scenario 1: Assume a > b (the larger number is greater than the smaller number)
- In this case, a^2 is greater than b^2 (since squaring a number makes it larger), so a^2 + b > b^2 + a.
Scenario 2: Assume a < b (the smaller number is greater than the larger number)
- In this case, b^2 is greater than a^2, so b^2 + a > a^2 + b.
Scenario 3: Assume a = b (the two numbers are equal)
- In this case, a^2 is equal to b^2, so a^2 + b = b^2 + a.
So, what will be true depends on the relationship between the larger number (a) and the smaller number (b). If a is greater than b, then a^2 + b > b^2 + a. If b is greater than a, then b^2 + a > a^2 + b. If a and b are equal, then both expressions are equal.