Lesson 9 iNumbers Develop Understanding

Ready

Solve each system of equations.

1.

${\begin{matrix} y = 3 x \\ y = - 2 x - 15 \end{matrix}$

2.

${\begin{matrix} 3 x + y = 21 \\ y = - 2 x - 15 \end{matrix}$

3.

${\begin{matrix} 3 x + 2 y = 7 \\ x - 2 y = - 3 \end{matrix}$

4.

${\begin{matrix} 5 x - y = 13 \\ - 2 x + y = - 1 \end{matrix}$

5.

${\begin{matrix} 3 x + y = 21 \\ - 3 x + 5 y = - 3 \end{matrix}$

6.

${\begin{matrix} 3 x + 2 y = 7 \\ x + y = 2 \end{matrix}$

Set

Perform the indicated operations. Then identify whether your answer is rational and real, irrational and real, or complex.

7.

$3 \sqrt{5} + 9 \sqrt{5}$

8.

$(16 + 9 \sqrt{5}) - (10 + 9 \sqrt{5})$

9.

$(4 \sqrt{3}) \cdot 2 \sqrt{3}$

10.

$(18 - 6 i) + (- 9 + 6 i)$

11.

$(18 - 4 i) + (7 + 21 i)$

12.

$(4 - 5 i) (- 1 + i)$

13.

$(6 - 8 i) (6 + 8 i)$

14.

$\sqrt{7} i \cdot \sqrt{2} i$

Indicate if the statement is true or false. If it is false, find a counter-example to prove the claim is false. ( Recall that a set of number is closed under an operation if it will always produce another number in the same set.)

15.

The set of irrational numbers is closed under addition.

16.

The set of integers is closed under addition and multiplication.

17.

The set of irrational numbers is closed under multiplication.

Go

Find the product of each pair of conjugates.

18.

$(x - 7) (x + 7)$

19.

$(2 x + 9) (2 x - 9)$

20.

$(5 x + 3) (5 x - 3)$

21.

$(a - b) (a + b)$

22.

$(8 + \sqrt{5}) (8 - \sqrt{5})$

23.

$(2 - \sqrt{7}) (2 + \sqrt{7})$

24.

$(12 + i \sqrt{39}) (12 - i \sqrt{39})$

25.

$(9 - i \sqrt{79}) (9 + i \sqrt{79})$

26.

$(x - 8 - \sqrt{5}) (x - 8 + \sqrt{5})$

27.

$(x - 9 + i \sqrt{79}) (x - 9 - i \sqrt{79})$

Find the two complex solutions. Recall that: $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ .

28.

$x^{2} - x + 2 = 0$

29.

$2 x^{2} + 3 x + 3 = 0$

30.

$x^{2} - 3 x + 3 = 0$

31.

$3 x^{2} - 3 x + 1 = 0$