<p>If <i>x</i> = 1 and <i>x</i> = 2 are solutions of equations \(x^3 + ax^2 + bx + c = 0\) and \(a + b = 1\), then find the value of <i>b</i>.</p>
Step-by-Step Solution
Key Concept: Since x = 1 and x = 2 are solutions of the cubic equation, they must satisfy it. Substituting these values creates a system of equations that, combined with the constraint a + b = 1, uniquely determines b.
<p><strong>Step 1:</strong> Since x = 1 is a solution:</p><p>1 + a + b + c = 0</p><p>∴ a + b + c = -1 ... (i)</p><p><strong>Step 2:</strong> Since x = 2 is a solution:</p><p>8 + 4a + 2b + c = 0</p><p>∴ 4a + 2b + c = -8 ... (ii)</p><p><strong>Step 3:</strong> We are given:</p><p>a + b = 1 ... (iii)</p><p><strong>Step 4:</strong> Subtract equation (i) from equation (ii):</p><p>(4a + 2b + c) - (a + b + c) = -8 - (-1)</p><p>3a + b = -7 ... (iv)</p><p><strong>Step 5:</strong> Subtract equation (iii) from equation (iv):</p><p>(3a + b) - (a + b) = -7 - 1</p><p>2a = -8</p><p>a = -4</p><p><strong>Step 6:</strong> Substitute a = -4 into equation (iii):</p><p>-4 + b = 1</p><p>b = 5</p><p>∴ <strong>Answer: b = 5</strong></p>
Correct Answer: b = 5