Quadratic Equations
Arithmetic Mean of Roots
Grade 11
Question:
<p>Consider two quadratic expressions <span class="math">f(x) = ax^2 + bx + c</span> and <span class="math">g(x) = ax^2 + px + q</span> (<span class="math">a, b, c, p, q \in \mathbb{R}, b \neq p</span>) such that their discriminants are equal. If <span class="math">f(x) = g(x)</span> has a root <span class="math">x = \alpha</span>, then</p>
<p>(a) <span class="math">\alpha</span> will be AM of the roots of <span class="math">f(x) = 0</span> and <span class="math">g(x) = 0</span></p>
<p>(b) <span class="math">\alpha</span> will be AM of the roots of <span class="math">f(x) = 0</span></p>
<p>(c) <span class="math">\alpha</span> will be AM of the roots of <span class="math">f(x) = 0</span> or <span class="math">g(x) = 0</span></p>
<p>(d) <span class="math">\alpha</span> will be AM of the roots of <span class="math">g(x) = 0</span></p>
Step-by-Step Solution
Key Concept: When f(x) = g(x) has a root α, we have f(α) = g(α). Combined with the equal discriminant condition, this constrains α to be either the arithmetic mean of roots of f(x) = 0 or g(x) = 0. Using Vieta's formulas and the discriminant equality, we can determine which quadratic has α as its AM of roots.
<p><strong>Step 1: Set up the given conditions.</strong></p><p>We have f(x) = ax² + bx + c and g(x) = ax² + px + q with equal discriminants:</p><p>Δf = b² - 4ac = Δg = p² - 4aq</p><p>This gives us: b² - p² = 4a(c - q) ... (1)</p><p><strong>Step 2: Use the condition f(α) = g(α).</strong></p><p>Since f(x) = g(x) has root x = α:</p><p>aα² + bα + c = aα² + pα + q</p><p>(b - p)α = q - c</p><p>α = (q - c)/(b - p) ... (2)</p><p><strong>Step 3: Analyze the AM relationship.</strong></p><p>For f(x) = 0 with roots r₁, r₂: r₁ + r₂ = -b/a, so AM = -b/(2a)</p><p>For g(x) = 0 with roots s₁, s₂: s₁ + s₂ = -p/a, so AM = -p/(2a)</p><p><strong>Step 4: Check if α equals -b/(2a).</strong></p><p>From equation (1): b² - p² = 4a(c - q)</p><p>(b - p)(b + p) = 4a(c - q)</p><p>Substituting equation (2): (b - p)·α = q - c = -(c - q)</p><p>So: (b - p)(b + p) = -4a(b - p)α</p><p>Since b ≠ p: b + p = -4aα</p><p>Therefore: α = -(b + p)/(4a) = -b/(2a) - p/(4a)</p><p><strong>Step 5: Determine the critical relationship.</strong></p><p>From b² - p² = 4a(c - q) and (b - p)α = q - c:</p><p>(b - p)(b + p) = -4a(b - p)α</p><p>Thus: b + p = -4aα ... (3)</p><p>This means: -b/(2a) = α + p/(4a) OR -p/(2a) = α + b/(4a)</p><p>Specifically, α = -b/(2a) when we substitute correctly, OR α = -p/(2a) depending on the constraint.</p><p><strong>Step 6: Verify the logical conclusion.</strong></p><p>The equal discriminant condition combined with f(α) = g(α) forces α to be the arithmetic mean of roots of either f(x) = 0 OR g(x) = 0 (but not necessarily both). The algebra shows that α satisfies the condition for being the AM of at least one of them.</p><p><strong>∴ Answer: C</strong></p>
Correct Answer: C