Quadratic Equations
Quadratic Equations
Allen Star Batch
Grade 11
Question:
Let $a = 0, b, c$ be integers and $\sin\theta, \cos\theta$ be the rational roots of the equation $ax^2 + bx + c = 0$. Then:
$a + 2c$ is a perfect square
$a + 2c$ is a perfect square
$a - 2c$ is a perfect square
$b$ is a perfect square
Step-by-Step Solution
Key Concept: The Pythagorean identity combined with Vieta's formulas imposes strong constraints that force certain expressions to be perfect squares.
Using $\sin \theta \cos \theta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ and the constraint $\sin^2\theta + \cos^2\theta = 1$, show that $b^2 - 2ac = a^2$ and $b^2 - 4ac = a(a-2c)$, proving that $a$, $a-2c$, and $a+2c$ are all perfect squares.
Correct Answer: 1,2,3