91Ó°ÊÓ

Solve compound inequality. \(3 \leq 4 x-3<19\)

The mathematician Girolamo Cardano is credited with the first use (in 1545 ) of negative square roots in solving the now-famous problem, "Find two numbers whose sum is 10 and whose product is \(40 .^{\prime \prime}\) Show that the complex numbers \(5+i \sqrt{15}\) and \(5-i \sqrt{15}\) satisfy the conditions of the problem. (Cardano did not use the symbolism \(i \sqrt{15}\) or even \(\sqrt{-15} .\) He wrote \(\mathrm{Rm} 15\) for \(\sqrt{-15},\) meaning "radix minus 15." He regarded the numbers \(5+\) R.m 15 and \(5-\) R.m 15 as "fictitious" or "ghost numbers," and considered the problem "manifestly impossible." But in a mathematically adventurous spirit, he exclaimed, "Nevertheless, we will operate."

Solve each equation in Exercises \(47-64\) by completing the square. $$x^{2}+3 x-1=0$$

Solve each equation by making an appropriate substitution. $$\left(x^{2}-x\right)^{2}-14\left(x^{2}-x\right)+24-0$$

Solve compound inequality. \(-3 \leq \frac{2}{3} x-5<-1\)

Solve compound inequality. \(-6 \leq \frac{1}{2} x-4<-3\)

Solve each equation by making an appropriate substitution. $$\left(x^{2}-2 x\right)^{2}-11\left(x^{2}-2 x\right)+24-0$$

Solve each equation in Exercises \(47-64\) by completing the square. $$x^{2}-3 x-5=0$$

Solve each equation in Exercises \(47-64\) by completing the square. $$2 x^{2}-7 x+3=0$$

Explain how to add complex numbers. Provide an example with your explanation.