The solution of $4^{x+1} + 4^{1-x} = 10$ is

Explanation

$$ 4^{x+1} + 4^{1-x} = 10 $$ $$ 4^x . 4 + 4 . 4^{-x} = 10 $$ $$ 4.4^x + \frac{4}{4^x} = 10 $$ $$ 4.(4^x)^2 + 4 - 10 \times 4^x = 0 $$ Let $4^x = z$ $$ 4z^2 + 4 - 10z = 0 $$ $$ 2z^2 - 5z + z = 0 $$ $$ 2z^2 - 4z - z + 2 = 0 $$ (490) ## Page 39 ## Layout Attribution (Critical) A table of layout regions detected on this page is provided inside <layout_regions> tags. Each region has an id, a label (e.g. text, table, image, header, footer, chart, etc.), and a bounding box in box_2d format [y_min, x_min, y_max, x_max] normalized to 0..1000 with top-left origin. Your task: before EVERY markdown element you output (paragraph, heading, table, list, code block, mermaid block, image placeholder, equation, page_header, page_footer, etc.), insert an HTML comment listing the layout region id(s) that correspond to that element: ``` ``` Rules: - Every markdown element MUST be preceded by a comment. - A single element may map to multiple layout regions (e.g. a table spanning two detected regions). List all matching ids separated by spaces. - A layout region should appear in at most one comment. - Use the bounding box and label to match visual regions on the screenshot to the content you are transcribing. - If a visible region has no matching layout id, you can omit the attribution comment for that element. - The layout region table is a reference for attribution only — the OCR text content is provided separately. Tables: - Output tables in HTML when possible. - Table should not be use for…