Quadratic Radicals
Name: _____________________
Score: _____________________
Part A: Multiple Choice (8 questions in total, 5 points each, 40 points in total)
1. Which of the following equation is correct?
A. !
!" !
#
=
"√!
%
B. !& =
#
'√()
&
*+ ! ,
(
D. !
C. !% = 4√2
)
=
+√*,
'
2. Calculate (2√5 − 2√3)(√12 + √20), what is the result?
A. 32
B. 16
C. 8
D. 4
3. In the following equations:
a) 3√3𝑎 + √27𝑎 = 6√3𝑎
b)
5√𝑥 − 2√𝑥 = 3√𝑥
c) √2 + √8 = √10
d)
√%-√#.
Which one is correct?
A. (a), (b), (d)
B. (a) and (c)
'
= √4 + √25 = 7
C. (a) and (b)
D. (b)
4. As the graph shown, real numbers 𝑎, 𝑏 are on the number line. What is the answer to
simplify this algebraic expression √𝑎' − √𝑏 ' − 5(𝑎 − 𝑏)' ?
A. −2𝑏
B. −2𝑎
C. 2(𝑏 − 𝑎)
D. 0
5. In the following radical expressions
"
√3，√𝑎 − 1，5𝑥 ' − 2𝑥 + 1, 5(−𝑥)* , √−4𝑥(𝑥 < 0), 5(𝑥 + 9)' − 1
How many of them are quadratic radicals?
A. Two
B. Three
C. Four
D. Five
6. If 5(4𝑎 − 1)' = 1 − 4𝑎, then
(
A. 𝑎 < )
(
B. 𝑎 ≤ )
(
C. 𝑎 > )
(
D. 𝑎 ≥ )
7. Which of the following best describes the domains for 𝑥, 𝑦 in this radical expression
'"
!/ ?
A. 𝑥 ≥ 0, 𝑦 ≠ 0
B. 𝑥 ≠ 0, 𝑦 ≠ 0
C. 𝑥, 𝑦 𝑎𝑟𝑒 𝑎𝑛𝑦 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠
D.
"
/
≥ 0, 𝑦 ≠ 0
8. If √3 − 𝑥 is a real number, then which of the following radical expression is NOT a
quadratic radical?
A. 5|𝑥 + 4|
B. √5 − 𝑥
C. 5−(𝑥 − 4)'
D. 5(𝑥 − 3)(𝑥 − 4)
Part B: Fill in the Blanks (7 questions in total, 5 points each, 35 points in total)
(
1. In this radical expression √2𝑥 + 3 +
, the domain for 𝑥 is _______________
"-(
2. When 𝑥 = √23 − 1, the value for 𝑥 ' + 2𝑥 + 2 =________________
(
3. In Physics, the equation for calculating an object free fall is 𝑆 = ' 𝑔𝑡 ' (𝑔 is the
gravitational acceleration, 𝑔 = 9.8 𝑚/𝑠 ' ). If the initial height is 88.2 𝑚, the time the
object falling to the ground is ___________________ (Simplify completely)
4. Given |𝑎| = −𝑎, simplify |1 − 𝑎| + 5(𝑎 − 2)' + 2𝑎 =______________
5. Observe this sequence: 0, √3, √6, 3, 2√3, √15, … (0 is the first number)
The 10th number is ___________ (Simplify completely)
6. If √3𝑥 + 5 and 2√2𝑥 + 7 are like terms and they are in the simplest form, then
𝑥 =___________
7. Simplify completely (√−𝑥)' + √𝑥 ' =_________________
Part C: Free Responses (3 questions in total)
1. (5 points each, 10 points in total)
Calculate and Simplify Completely (You must show your work to earn full credits)
(
(
(1) L√0.5 − 2!*M − (!% − √75)
(2) Given 𝑥 = √3 + 1, 𝑦 = √3 − 1, calculate and simplify
" ! 0'"/-/ !
" ! 0/ !
2. (15 points) As the graph shown, the larger square with area 8𝑥 is composed of a smaller
square with area 𝑥 and four congruent rectangles. Find the perimeter of the rectangle.
Express your answer in terms of 𝑥.