Question

Find the discriminant and tell how many solutions exist: -8x^2 - 10x - 5 = 2

Answer 1

-8x² - 10x - 7 = 0

8x² + 10x + 7 = 0

10² - 4(8)(7) = 100 - 224 = -124

This equation has no real solutions (two complex solutions).

Answer 2

`-8x^2-10x-5=2\implies -8x^2-10x-7=0 \\\\[-0.35em] ~\dotfill\\\\ ~~ \hspace{5em} \textit{discriminant of a quadratic}`

`\stackrel{\stackrel{\textit{\small a}}{\downarrow }}{-8}x^2 \stackrel{\stackrel{\textit{\small b}}{\downarrow }}{-10}x \stackrel{\stackrel{\textit{\small c}}{\downarrow }}{-7}=0 \hfill \stackrel{discriminant}{b^2-4ac}~~ = ~~ \stackrel{ \textit{\large \underline{Number of Solutions}} }{\begin{cases} ~~ ~~ 0&\textit{one solution}\\ positive&\textit{two solutions}\\ negative&\textit{no solution}\\ \end{cases}} \\\\\\ (-10)^2-4(-8)(-7) \implies 100 - 224 \implies -124 ~~ \textit{no solution}`