Quadratic Equations
Nature of roots
Grade 11
Question:
<p>Let \(a, b, c \in \mathbb{Q}^+\) satisfying \(a > b > c\). Which of the following statement(s) hold true for the quadratic polynomial \(f(x) = (a+b-2c)x^2 + (b+c-2a)x + (c+a-2b)\)?</p><ol><li>The mouth of the parabola \(y = f(x)\) opens upwards</li><li>Both roots of the equation \(f(x) = 0\) are rational</li><li>The \(x\)-coordinate of vertex of the graph is positive</li><li>The product of the roots is always negative</li></ol>
<p>The mouth of the parabola \(y = f(x)\) opens upwards</p>
<p>Both roots of the equation \(f(x) = 0\) are rational</p>
<p>The \(x\)-coordinate of vertex of the graph is positive</p>
<p>The product of the roots is always negative</p>
Step-by-Step Solution
Key Concept: Recognize that the coefficients have a cyclic structure where each coefficient equals the sum of two variables minus twice the third. The constraint a > b > c combined with the coefficient structure determines the sign of the leading coefficient and discriminant properties.
<p><strong>Step 1: Analyze the leading coefficient</strong></p><p>Leading coefficient: <code>A = a + b - 2c</code></p><p>Since a > b > c, we have a + b > 2c (as a > c and b > c implies a + b > 2c). Thus <code>A > 0</code> ✓ <strong>Statement A is TRUE</strong></p><p><strong>Step 2: Check the discriminant for rationality of roots</strong></p><p>Coefficient B = b + c - 2a, Coefficient C = c + a - 2b</p><p>Discriminant: <code>Δ = B² - 4AC = (b+c-2a)² - 4(a+b-2c)(c+a-2b)</code></p><p>Expanding and simplifying (key observation): <code>Δ = (b+c-2a)² - 4(a+b-2c)(c+a-2b)</code></p><p>After algebraic manipulation: <code>Δ = (a-b)² + (b-c)² + (c-a)² > 0</code> (perfect square sum)</p><p>Moreover, this discriminant is a perfect square of a rational expression, making both roots rational. ✓ <strong>Statement B is TRUE</strong></p><p><strong>Step 3: Analyze the x-coordinate of vertex</strong></p><p>Vertex x-coordinate: <code>x = -B/(2A) = -(b+c-2a)/(2(a+b-2c)) = (2a-b-c)/(2(a+b-2c))</code></p><p>Since a > b > c: numerator = 2a - b - c > 0 (as 2a > a + b and a > c)</p><p>Denominator a + b - 2c > 0 (from Step 1)</p><p>Therefore <code>x-coordinate > 0</code> ✓ <strong>Statement C is TRUE</strong></p><p><strong>Step 4: Analyze product of roots</strong></p><p>Product of roots: <code>C/A = (c+a-2b)/(a+b-2c)</code></p><p>Since b is between a and c with a > b > c, we have: c + a - 2b can be positive or negative depending on the specific values (e.g., if a = 5, b = 3, c = 1: product = 0/4 = 0; if a = 10, b = 3, c = 1: product is positive). The sign is not always negative. ✗ <strong>Statement D is FALSE</strong></p><p><strong>∴ Answer: A, B, C</strong></p>
Correct Answer: A,B,C