Introduction / Context:
This algebra problem focuses on symmetric expressions in two variables, namely sums and products of a and b. Given a + b and ab, we can find a² + b² without solving for a and b individually by using the formula for the square of a binomial. This kind of manipulation is common in quadratic equations and identities.
Given Data / Assumptions:
- a + b = 5.
- ab = 3.
- We are asked to find a² + b².
Concept / Approach:
Key concepts:
- Use the identity (a + b)² = a² + 2ab + b².
- Rearrange this identity to express a² + b² in terms of a + b and ab.
- Substitute the given numerical values to compute the result directly.
Step-by-Step Solution:
Step 1: Recall the identity (a + b)² = a² + 2ab + b².
Step 2: Rearrange to express a² + b²: a² + b² = (a + b)² − 2ab.
Step 3: Substitute a + b = 5: (a + b)² = 5² = 25.
Step 4: Substitute ab = 3: 2ab = 2 × 3 = 6.
Step 5: Compute a² + b² = 25 − 6 = 19.
Verification / Alternative check:
Step 1: One can construct the quadratic equation whose roots are a and b: t² − (a + b)t + ab = 0, so t² − 5t + 3 = 0.
Step 2: Solve this quadratic to find explicit values of a and b using the quadratic formula.
Step 3: After obtaining numeric values for a and b, compute a² + b² directly, which will simplify numerically to 19, confirming the identity based approach.
Why Other Options Are Wrong:
Option 17 would result from incorrectly computing 25 − 8 instead of 25 − 6.
Option 18 may appear if someone uses 2ab = 7 by mistake or misreads the problem data.
Option 20 could result from adding instead of subtracting, that is, taking 25 + (−5) incorrectly.
Option 16 does not correspond to any correct manipulation of the given relationships.
Common Pitfalls:
Students sometimes misremember the identity and write (a + b)² = a² + b² instead of including the 2ab term.
Another error is to confuse ab with a²b² or to square ab accidentally while manipulating.
Some learners try to solve for a and b directly, which is unnecessary work when symmetric identities give the answer quickly.
Final Answer:
The value of a² + b² is
19.
Discussion & Comments