All the roots of \[x^2 + px + q = 0\]are real, where $p$ and $q$ are real numbers. Prove that all the roots of
\[x^2 + px + q + (x + a)(2x + p) = 0\]are real, for any real number $a.$

Aperpendicular bisector, cd is drawn through point con ab if the coordinates of point a are (-3, 2) and the coordinates of point b are (7,6), the x-intercept of cd is point lies on cd.

After reading 80% of her e-mails in her inbox, danette still has m unread e-mails. which of the following expressions could represent the number of e-mails danette had in her inbox before she started reading? choose 2 answers: (choice a) 5m (choice b) 1.8m (choice c) 80m (choice d) m/0.8 (choice e) m/1−0.8

Use mathematical induction to prove the statement is true for all positive integers n, or show why it is false. 1^2 + 4^2 + 7^2 + + (3n - 2)^2 = [n(6n^2-3n-1)/2]