91Ó°ÊÓ

The discriminant of the equation \(a x^{2}+b x+c=0\) (with \(a, b, c\) integers) is given. Use it to determine whether or not the solutions of the equation are rational numbers. $$b^{2}-4 a c=25$$

The discriminant of the equation \(a x^{2}+b x+c=0\) (with \(a, b, c\) integers) is given. Use it to determine whether or not the solutions of the equation are rational numbers. $$b^{2}-4 a c=72$$

Suppose \(a, b, c\) are fixed real numbers such that \(b^{2}-4 a c \geq 0 .\) Let \(r\) and \(s\) be the solutions of $$ a x^{2}+b x+c=0 $$ (a) Use the quadratic formula to show that \(r+s=-b / a\) and \(r s=c / a\) (b) Use part (a) to verify that \(a x^{2}+b x+c=\) \(a(x-r)(x-s)\) (c) Use part (b) to factor \(x^{2}-2 x-1\) and \(5 x^{2}+8 x+2\)

Express the given geometric statement about numbers on the number line algebraically, using absolute values. is more than 6 units from \(c\)

Use the geometric approach explained in the text to solve the given equation or inequality. $$|x-5|<2$$