Quadratic Equations
Quadratic Equations
Allen Star Batch
Grade 11
Question:
Consider the equation $\sqrt{x + \sqrt{2x - 1}} + \sqrt{x - \sqrt{2x - 1}} = A$
For $A = \sqrt{2}$, $x \in [\frac{1}{2}, 1]$
For $A = \sqrt{2}$, $x \in [0, \frac{1}{2}]$
For $A = 1$, $x \in \phi$
For $A = 2$, $x = \frac{3}{2}$
Step-by-Step Solution
Key Concept: Combine multiple inequalities involving the same variables, then use algebraic identities and distinctness conditions to constrain the ratio.
Let $\sqrt{2x-1}=t$, transforming the equation to $|t+1|+|t-1|=4\sqrt{2}$. By considering cases for $t\leq-1$, $-10, b^2\leq 4ac$; $b>0, c^2\leq 4ab$; and $c>0, a^2\leq 4bc$, summing these yields $a^2+b^2+c^2ab+bc+ca$, so $\frac{a^2+b^2+c^2}{ab+bc+ca}\in(1,4)$.
Correct Answer: 1,3,4