Chapter 3

Sketch a graph of \(f(x)=a x^{2}+b x+c\) that satisfies each set of conditions. $$a<0, b^{2}-4 a c>0$$

Show that \(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2} i\) is a square root of \(i\)

Sketch a graph of \(f(x)=a x^{2}+b x+c\) that satisfies each set of conditions. $$a>0, b^{2}-4 a c>0$$

Find the complex conjugate. $$5-3 i$$

Sketch a graph of \(f(x)=a x^{2}+b x+c\) that satisfies each set of conditions. $$a>0, b^{2}-4 a c=0$$

Find the complex conjugate. $$-3+i$$

Find the complex conjugate. $$-18 i$$

Find the complex conjugate. $$8 i$$

Find the complex conjugate. $$-\sqrt{8}$$

Divide as indicated. Write each quotient in standand form. $$\frac{9+2 i}{2+i}$$

Divide as indicated. Write each quotient in standand form. $$\frac{16+2 i}{3+i}$$

Find the complex conjugate. $$\frac{-11-7 i}{1+2 i}$$

Divide as indicated. Write each quotient in standard form. \(\frac{6+22 i}{1-3 i}\)

Find the complex conjugate. $$\frac{3}{-2 i}$$

Find the complex conjugate. $$\frac{7}{-4 i}$$

Find the complex conjugate. $$\frac{-3+4 i}{2-i}$$

Find the complex conjugate. $$\frac{-6+8 i}{1-i}$$

Find the complex conjugate. $$\frac{4-3 i}{4+3 i}$$

Find the complex conjugate. $$\frac{2-i}{2+i}$$

Solve each inequality analytically. Support your answers graphically. Give exact values for endpoints. (a) \(x^{2}+4 x+3 \geq 0\) (b) \(x^{2}+4 x+3<0\)

Find the complex conjugate. $$\frac{-31-6 i}{i}$$

Solve each inequality analytically. Support your answers graphically. Give exact values for endpoints. (a) \(x^{2}+6 x+8<0\) (b) \(x^{2}+6 x+8 \geq 0\)

Find the complex conjugate. \(\frac{-19-9 i}{i}\)

Solve each inequality analytically. Support your answers graphically. Give exact values for endpoints. (a) \(2 x^{2}-9 x>-4\) (b) \(2 x^{2}-9 x \leq-4\)

Explain why the method of dividing complex numbers (that is, multiplying both the numerator and the denominator by the conjugate of the denominator) works. What property justifies this process?

Solve each inequality analytically. Support your answers graphically. Give exact values for endpoints. (a) \(3 x^{2}+13 x \leq-10\) (b) \(3 x^{2}+13 x>-10\)

Suppose that your friend describes a method of simplifying a positive power of \(i .\) "Just divide the exponent by 4 and then look at the remainder. Then, refer to the short table of powers of \(i\) in this section. The given power of \(i\) is equal to \(i\) to the power indicated by the remainder. And if the remainder is \(0,\) the result is \(i^{0}=1 .^{\prime \prime}\) Explain why this method works.

Solve each inequality analytically. Support your answers graphically. Give exact values for endpoints. (a) \(-x^{2}-x \leq 0\) (b) \(-x^{2}-x>0\)

Solve each inequality analytically. Support your answers graphically. Give exact values for endpoints. (a) \(-x^{2}+2 x \leq 0\) (b) \(-x^{2}+2 x>0\)

Solve each inequality analytically. Support your answers graphically. Give exact values for endpoints. (a) \(2 x^{2}-x+3<0\) (b) \(2 x^{2}-x+3 \geq 0\)

Solve each inequality analytically. Support your answers graphically. Give exact values for endpoints. (a) \(2 x+1 \geq x^{2}\) (b) \(2 x+1

Solve each inequality analytically. Support your answers graphically. Give exact values for endpoints. (a) \(x^{2}+5 x<2\) (b) \(x^{2}+5 x \geq 2\)

Solve each inequality analytically. Support your answers graphically. Give exact values for endpoints. (a) \(x-3 x^{2}>-1\) (b) \(x-3 x^{2} \leq-1\)

Solve each inequality analytically. Support your answers graphically. Give exact values for endpoints. (a) \(-2 x^{2}+3 x<-4\) (b) \(-2 x^{2}+3 x \geq-4\)

Solve each formula for the indicated variable. Leave \(\pm\) in answers when applicable. Assume that no denominators are 0 $$s=\frac{1}{2} g t^{2} \quad \text { for } t$$

Solve each formula for the indicated variable. Leave \(\pm\) in answers when applicable. Assume that no denominators are 0 $$s l=\pi r^{2} \text { for } r$$

Solve each formula for the indicated variable. Leave \(\pm\) in answers when applicable. Assume that no denominators are 0 $$a^{2}+b^{2}=c^{2} \quad \text { for } a$$

Solve each formula for the indicated variable. Leave \(\pm\) in answers when applicable. Assume that no denominators are 0 $$x=s^{2} \quad \text { for } s$$

Solve each formula for the indicated variable. Leave \(\pm\) in answers when applicable. Assume that no denominators are 0 $$S=4 \pi r^{2} \quad \text { for } r$$

Solve each formula for the indicated variable. Leave \(\pm\) in answers when applicable. Assume that no denominators are 0 $$V=\frac{1}{3} \pi r^{2} h \quad \text { for } r$$

Solve each formula for the indicated variable. Leave \(\pm\) in answers when applicable. Assume that no denominators are 0 $$V=e^{3} \quad \text { for } e$$

Solve each formula for the indicated variable. Leave \(\pm\) in answers when applicable. Assume that no denominators are 0 $$V=\frac{4}{3} \pi r^{3} \quad \text { for } r$$

Solve each formula for the indicated variable. Leave \(\pm\) in answers when applicable. Assume that no denominators are 0 $$F=\frac{k M v^{4}}{r} \text { for } v$$

Solve each formula for the indicated variable. Leave \(\pm\) in answers when applicable. Assume that no denominators are 0 $$s=s_{0}+g t^{2}+k \quad \text { for } t$$

Solve each formula for the indicated variable. Leave \(\pm\) in answers when applicable. Assume that no denominators are 0 $$P=\frac{E^{2} R}{(r+R)^{2}} \text { for } R$$

Solve each formula for the indicated variable. Leave \(\pm\) in answers when applicable. Assume that no denominators are 0 $$S=2 \pi r h+2 \pi r^{2} \quad \text { for } r$$

Solve each equation for \(x\) and then for \(y .\) $$x^{2}+x y+y^{2}=0 \quad(x>0, y>0)$$

Find the values of \(b\) for which the equation \(x^{2}+b x+4=0\) has two real solutions.