Download Worksheet 3.1

Algebra 2 Unit 2
Worksheet 3.1
Name __________________________________
Part 1: Square roots and imaginary numbers
Simplifying square roots Perfect Squares: If the number under the radical sign has identical factors, it is called a perfect square. If the number under a square root sign is a perfect square number (e.g. 25), take the square root and write the answer without the radical sign. This is the simplified answer. Example: 25 = 5 Non-­‐Perfect Squares: If the square root is NOT a perfect square (e.g. 24) simplify the square roots into 2 parts.First part: largest perfect square factor (rational root) Second part: a non-­‐perfect square factor (irrational root) • Simplify the perfect square root . Put that number OUTSIDE the radical sign. Note: it is STILL multiplied so it should be immediately to the left of the radical sign without any symbol after it. • Leave the irrational square root under the radical sign. Example: 24 = 4 i 6 = 4 i 6 = 2 6 1) Simplify the following: 60 a) 54 e) 160 i)
b)
80 f)
32 j)
13 c)
28 g)
175 k) 4 7 d)
100 h)
27 l)
4 8
Simplifying square roots (with imaginary numbers) • Negative Perfect Squares: If the number under a square root sign is a negative perfect square number (e.g. -­‐25), factor the ONE square root into two parts: • First part: largest perfect square factor (rational root) • Second part: −1 • Simplify the perfect square root and write the letter i next to it on the right. This will be your simplified answer. Example: −25 = −1i 25 = −1 i 25 = 5i Note that i comes AFTER the 5 •
Negative Non-­‐Perfect Squares: If the square root is NOT a perfect square (e.g. -­‐24) we will need to simplify the square roots into three parts, one of which is −1 . • First part: largest perfect square factor (rational root) • Second part: a non-­‐perfect square factor (irrational root) • Third part: −1 Simplify the perfect square root and write the letter i next to it on the right. Leave the irrational square root under the radical sign. 1
Example: −24 = −1i 4 i 6 = −1 i 4 i 6 = 2i 6 Note: i comes AFTER the 2 but BEFORE the square root sign. Product Property of
Square Roots
Quotient Property of
Square Roots
a
a
=
b
b
ab = a b
Definition of i
i = −1
2) Simplify the following square roots COMPLETELY. Use i in your answer if it is imaginary.
a) 250 b) 0 c) − 4 d) 6 − 4 e) − 48 f) − 48 g) − 3 50 h) − 28 i) 4 500 j) − 250 k) 3 ± −125 l) 3 ± −50 m) 3 + 49 n) −4 − 100 o) 3 ± −49 p) −5 ± 81 5
q) − 9
Answers (scrambled): −15 2 0 10 2i 7 5 10 3 ± 5i 5 4i 3 −4 3 4 and −14 2
−2 3 ± 7i −14 12i 3 ± 5i 2 i 5
3
5i 10 40 5
Part 2: Solving Quadratic Equations using Square Roots
To solve simple quadratic equations of the form x 2 = a, use the method of
taking square roots.
Example Solve 3x + 45 = 0
Example Solve x 2 − 28 = 0.
2
x 2 = 28
Add 28 to both sides.
Take the square root.
x 2 = 28
Simplify the square root.
3x 2 = −45
Divide by 3
x 2 = −15
Take the square root of both
sides.
x = ± 28
Note: the ± always comes
BEFORE the radical part.
Subtract 45 from both sides.
x = ± 4•7
Simplify the square root on the
side it remains. NOTE: the ±
comes BEFORE the i
x = ±2 7
x 2 = −15
x = ± −15
x = ±i 15
Note that there are always TWO answers to a quadratic equation. Here is a simple example to show why: x2 = 4
2
Solve x = 4 x = ±2
Solve each equation by taking square roots. Show your solving work. Simplify your
answers completely. Use a separate sheet of paper if you need more room but BE NEAT!
1.
4.
x 2 − 20 = 0 .
1 2
x − 3 = 5 2
3) −2 x 2 + 7 = −1
2) x 2 + 48 = 0
7) −5 = 2(x − 3)2 + 9 5. −4x 2 + 13 = 0
8) (5x + 4)2 + 9 = −40 6) (x − 10)2 + 8 = 12 2
9) (y + 10) − 49 = 0 Solutions (scrambled):
±4 3
12 and 8
±2 5
3± 7
±4 ±2
3
− 3 and − 17
-4 ± i 7
5
± 13
2
Part 3: Challenge problems:
1) Square each number below and simplify your answer completely. Hint: if
i = −1 then what would i 2 equal?
a. 4i ___________
b. i 11 ___________
c.
i 7
3 ___________
2) Solve each equation (give any imaginary solutions).
a. 7x 2 − 12 = 0
(
) (
c. 2 x 2 − 1 = x 2 − 3
b. x 2 + 9 = 3
)
3) Recall the equation for falling objects: h(t ) = h0 − 16t 2 , where h is the height of the object, in feet, at
any time t, in seconds, and h0 is the object’s initial height in feet. Use this equation for Problems 10–11.
a.A carpenter dropped a hammer from a rooftop 48 feet above ground. How long did it take the hammer to hit
the ground? Use the equation and show your work:
________________________________________________________________________________
b.
An acorn fell from a branch 20 feet high and landed on a branch 7 feet high.
How long did it take the acorn to fall? Use the equation and show your work:
________________________________________________________________________________
4