Which one of the following formulas correctly expresses this statement: A quantity [tex]$x$[/tex] is equal to the sum of the squares of [tex]$a$[/tex] and [tex]$b$[/tex]?
A. [tex]$x=2 a+2 b$[/tex]
B. [tex]$x=\sqrt{a+b}$[/tex]
C. [tex]$x=\sqrt{a b}$[/tex]
D. [tex]$x=a^2+b^2$[/tex]
Answer (2)
The problem states that x is the sum of the squares of a and b .
Express the sum of squares as a 2 + b 2 .
The formula that represents this statement is x = a 2 + b 2 .
The correct option is D: x = a 2 + b 2 .
Explanation
Understanding the Problem We are given the statement: A quantity x is equal to the sum of the squares of a and b . We need to find the formula that correctly represents this statement from the given options.
Expressing the Statement as a Formula The sum of the squares of a and b can be written as a 2 + b 2 . Therefore, the formula that correctly expresses the given statement is x = a 2 + b 2 .
Comparing with the Options Now, let's compare this formula with the given options:
A. x = 2 a + 2 b B. x = a + b C. x = ab D. x = a 2 + b 2
We can see that option D matches the formula we derived.
Final Answer Therefore, the correct formula is x = a 2 + b 2 .
Examples
In physics, the Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. If we denote the length of the hypotenuse as x and the lengths of the other two sides as a and b , then the theorem can be expressed as x 2 = a 2 + b 2 . This formula is a direct application of the concept of the sum of squares.
The correct formula that expresses the statement "A quantity x is equal to the sum of the squares of a and b " is x = a 2 + b 2 . Therefore, the selected answer is option D. This accurately represents the relationship described in the question.
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