<p><i>a</i> and <i>b</i> are roots of the equation \(2x^2 - 35x + 2 = 0\). Find the value of \((2a - 35)^3(2b - 35)^3\).</p>
Step-by-Step Solution
Key Concept: Since a and b are roots of 2x² - 35x + 2 = 0, we have 2a² - 35a + 2 = 0 and 2b² - 35b + 2 = 0. This means 2a - 35 = -2/a and 2b - 35 = -2/b, allowing us to simplify the given expression dramatically.
<p><strong>Step 1: Use the root condition</strong></p><p>Since a is a root of 2x² - 35x + 2 = 0:</p><p>2a² - 35a + 2 = 0</p><p>Rearranging: 2a² + 2 = 35a</p><p>Dividing by a: 2a + 2/a = 35</p><p>Therefore: 2a - 35 = -2/a</p><p><strong>Step 2: Apply the same logic to b</strong></p><p>Similarly, since b is a root:</p><p>2b - 35 = -2/b</p><p><strong>Step 3: Substitute into the given expression</strong></p><p>(2a - 35)³(2b - 35)³ = (-2/a)³(-2/b)³</p><p>= (-2)³/a³ × (-2)³/b³</p><p>= (-8)/a³ × (-8)/b³</p><p>= 64/(a³b³)</p><p>= 64/(ab)³</p><p><strong>Step 4: Find the product ab using Vieta's formulas</strong></p><p>For the equation 2x² - 35x + 2 = 0, the product of roots is:</p><p>ab = c/a_coefficient = 2/2 = 1</p><p><strong>Step 5: Calculate the final answer</strong></p><p>(2a - 35)³(2b - 35)³ = 64/(ab)³ = 64/(1)³ = 64</p><p><strong>∴ Answer: 64</strong></p>
Correct Answer: 64