If a + b + c = 9 and ab + bc + ac = 23 then what would be the value of the a^2 + b^2 + c^2?

Appeared in: SGPGI Nursing Officer-2023

Explanation

The problem is solved using the standard algebraic identity: (a + b + c)² = a² + b² + c² + 2(ab + bc + ac).
To find the required value, the formula is rearranged to: a² + b² + c² = (a + b + c)² - 2(ab + bc + ac).
The given values are a + b + c = 9 and ab + bc + ac = 23.
Substituting the values gives: a² + b² + c² = (9)² - 2(23).
Calculation: 81 - 46 = 35.

Option A: This is the value of the sum of the products of the variables taken two at a time (ab + bc + ac), not the sum of their squares.
Option B: This is the square of (a + b + c), which is 9². It represents only the first part of the formula, (a + b + c)², without subtracting the term 2(ab + bc + ac).
Option D: This is an incorrect value that arises from a calculation error during subtraction (e.g., 81 - 49). The correct calculation is 81 - 46.

Clinical Relevance

Nursing practice connection: This is primarily an exam-oriented knowledge point with limited direct bedside application, so retain Algebraic Identities as background academic context rather than a clinical decision trigger.
This is a general aptitude and reasoning question, not directly related to clinical nursing practice.
Such questions are included in nursing entrance exams to assess a candidate's basic mathematical and problem-solving skills, which are essential for tasks like dosage calculations and data interpretation.
What if the question asked for (a-b)² + (b-c)² + (c-a)²? You would use the identity 2(a² + b² + c² - ab - bc - ca), which would be 2(35 - 23) = 2(12) = 24.

How to Approach the Question

First, identify the knowns and the unknown. You are given (a + b + c) and (ab + bc + ac), and you need to find (a² + b² + c²).
Recall the relevant algebraic identity that connects these three expressions: (a + b + c)² = a² + b² + c² + 2(ab + bc + ac).
Rearrange the formula to isolate the term you need to find: a² + b² + c² = (a + b + c)² - 2(ab + bc + ac).
Substitute the given numerical values into the rearranged formula.
Perform the arithmetic calculation carefully: (9)² - 2(23) = 81 - 46.
The result of the calculation is the answer. Compare it with the given options.

Concept Tested & Keywords

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