Find two consecutive odd positive integers, sum of whose squares is $290$.
Step-by-Step Solution
Key Concept: Integers $x, x + 2$. $x^2 + (x + 2)^2 = 290 \Rightarrow 2x^2 + 4x + 4 = 290 \Rightarrow 2x^2 + 4x - 286 = 0 \Rightarrow x^2 + 2x - 143 = 0 \Rightarrow (x + 13)(x - 11) = 0 \Rightarrow x = 11$. Numbers are $11$ and $13$.
$x^2 + (x + 2)^2 = 290 \Rightarrow x^2 + 2x - 143 = 0$. [1.0 Mark]
$(x + 13)(x - 11) = 0 \Rightarrow x = 11$ (since positive). Integers are $11$ and $13$. [1.0 Mark]
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🎯 Official CBSE Marking Scheme:
Forming quadratic $x^2 + 2x - 143 = 0$: 1.0 Mark
Solving $x = 11$ and finding numbers $11, 13$: 1.0 Mark
Correct Answer: