Quadratic Equations
Quadratic Equations
Allen Star Batch
Grade 11
Question:
The equation $8x^4 - 16x^3 + 16x^2 - 8x + a = 0$, $a \in \mathbb{R}$ has:
Atleast two real roots $\forall a \in \mathbb{R}$
Atleast two imaginary roots $\forall a \in \mathbb{R}$
The sum of all non-real roots equal to $2$, if $a > \frac{3}{2}$
The sum of all non-real roots equal to $1$, if $a \leq \frac{3}{2}$
Step-by-Step Solution
Key Concept: Factor the polynomial and exploit symmetry properties to determine which horizontal lines $y=-\frac{a}{8}$ intersect the graph at exactly two points.
From $x^4-2x^3+2x^2-x=-\frac{a}{8}$, factor as $x(x-1)(x^2+1)=-\frac{a}{8}$. Analyze $f(x)=x(x-1)(x^2+1)$: it has $f'(\frac{1}{2})=0$ and $f(1-x)=f(x)$, showing symmetry about $x=\frac{1}{2}$. For the equation to have exactly two real roots, require $-\frac{a}{8}=-\frac{3}{16}$, giving $a=\frac{3}{2}$.
Correct Answer: 2,3,4