<p>Given <span class="math">f(x) = ax^4 - 2ax^2 + e</span>, if <span class="math">f(x) = f(0)</span>, find the sum of squares of all elements in the set <span class="math">T = \{x \in \mathbb{R} | f(x) = f(0)\}</span>.</p>
Step-by-Step Solution
Key Concept: Factor the equation by setting f(x) = f(0) and solve for all real roots, then sum their squares.
<p><strong>Step 1:</strong> Given <span class="math">f(x) = ax^4 - 2ax^2 + e</span> and <span class="math">f(x) = f(0)</span>.</p><p><strong>Step 2:</strong> Since <span class="math">f(0) = e</span>, we have:</p><p><span class="math">ax^4 - 2ax^2 + e = e</span></p><p><span class="math">ax^4 - 2ax^2 = 0</span></p><p><span class="math">ax^2(x^2 - 2) = 0</span></p><p><strong>Step 3:</strong> This gives us <span class="math">x = 0, 0, -\sqrt{2}, \sqrt{2}</span>.</p><p><strong>Step 4:</strong> Therefore, <span class="math">T = \{0, -\sqrt{2}, \sqrt{2}\}</span>.</p><p><strong>Step 5:</strong> Sum of squares of all elements:</p><p><span class="math">0^2 + (-\sqrt{2})^2 + (\sqrt{2})^2 = 0 + 2 + 2 = 4</span></p><p>∴ Answer is <strong>4</strong>.</p>
Correct Answer: 4