Range of values of the variable p is [tex]0\leq p\leq \frac{10}{3}[/tex]
Step-by-step explanation:
From general quadratic equation [tex]ax^{2} +bx+c[/tex], for imaginary case-
[tex]D\leq 0\\b^2-4ac\leq 0[/tex]
In given question, [tex]a= p+2[/tex], [tex]b=2p[/tex], [tex]c= 5-p[/tex]
using above condition-
[tex]b^2-4ac = 4p^2-4(2p)(5-p)\leq 0\\4p^2 - 40p+8p^2\leq 0\\12p^2-40p\leq 0\\4p(3p-10)\leq 0[/tex]
Here, [tex]4p\leq 0, p\leq 0[/tex]
and
[tex]3p-10\leq 0\\p\leq \frac{10}{3}[/tex]
Therefore, range of values of the variable p is [tex]0\leq p\leq \frac{10}{3}[/tex].
To learn more about real and imaginary roots of Quadratic equations-https://brainly.com/question/18659539
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