Question

Find the discriminant and State of real and imaginary solutions.

9n^2 - 3n - 8= -10

-2x^2 - 8x -14= -6​

Answer 1

`\qquad \qquad \qquad \textit{discriminant of a quadratic} \\\\\\ 9n^2-3n-8=-10\implies 9n^2-3n+2=0 \\\\\\ \stackrel{\stackrel{a}{\downarrow }}{9}n^2\stackrel{\stackrel{b}{\downarrow }}{-3}n\stackrel{\stackrel{c}{\downarrow }}{+2}=0 ~~~~~~~~ \stackrel{discriminant}{b^2-4ac}= \begin{cases} 0&\textit{one solution}\\ positive&\textit{two solutions}\\ negative&\textit{no solution} \end{cases}`

`(-3)^2-4(9)(2)\implies 9-72\implies -63\leftarrow \stackrel{\textit{two \underline{imaginary} solutions}}{\textit{no real solution}} \\\\[-0.35em] ~\dotfill\\\\ -2x^2-8x-14=-6\implies -2x^2-8x-8=0 \\\\\\ \stackrel{\stackrel{a}{\downarrow }}{-2}x^2\stackrel{\stackrel{b}{\downarrow }}{-8}x\stackrel{\stackrel{-8}{\downarrow }}{-8}=0\qquad \qquad \qquad \qquad (-8)^2-4(-2)(-8) \\\\\\ 64-64\implies 0\leftarrow \textit{one \underline{real} solution}`