Quadratic Equations
System of Equations with Roots
Grade 11
Question:
<p>If x and y are positive integers such that <span>xy + x + y = 71</span> and <span>x<sup>2</sup>y + xy<sup>2</sup> = 880</span>, then <span>x<sup>2</sup> + y<sup>2</sup></span> is equal to</p>
<p>(a) 125</p>
<p>(b) 137</p>
<p>(c) 146</p>
<p>(d) 152</p>
Step-by-Step Solution
Key Concept: Convert the system into symmetric functions by treating xy and (x+y) as roots of a quadratic equation.
<p><strong>Step 1:</strong> From the given equations:</p><p><span>xy + (x + y) = 71</span></p><p><span>xy(x + y) = 880</span></p><p><strong>Step 2:</strong> Let <span>s = x + y</span> and <span>p = xy</span>. Then:</p><p><span>p + s = 71</span></p><p><span>ps = 880</span></p><p><strong>Step 3:</strong> Thus p and s are roots of: <span>t<sup>2</sup> - 71t + 880 = 0</span></p><p><strong>Step 4:</strong> Factoring: <span>(t - 55)(t - 16) = 0</span></p><p>So <span>t = 55</span> or <span>t = 16</span></p><p><strong>Step 5:</strong> If <span>xy = 55</span> and <span>x + y = 16</span> (taking the valid case for positive integers):</p><p><span>x<sup>2</sup> + y<sup>2</sup> = (x + y)<sup>2</sup> - 2xy = 16<sup>2</sup> - 2(55) = 256 - 110 = 146</span></p><p>∴ Answer is (c).</p>
Correct Answer: C