RANDOM TUTORIAL FOR JUNIOR HIGH SCHOOL:
1.) Evaluate: (1/3 + 1/8)² - (1/3 - 1/8)²
Answer: 1/6
Solution #1
1/3 + 1/8 = 11/24
1/3 - 1/8 = 5/24
(11/24)² - (5/24)² = 121/576 - 25/576
96/576 = 1/6
Solution #2
Let a = 1/3 and b = 1/8
(a + b)² - (a - b)²
(a² + 2ab + b²) - (a² -2ab + b²) = 4ab
Thus, 4(1/3)(1/8) = 1/6
2.) Express 0.1696969... into a fraction in lowest term
Answer: 28/165
Solution:
Let x = 0.1696969...
1000x = 169.696969...
10x = 1.696969...
Deduct the two equations
990x = 168
x = 168/990
x = 28/165
3.) After an hour of rain the temperature outdoors decreased by 11°C. By how many °F did the temperature decreased?
Answer: 19.8
Solution #1:
Choose any starting temperature (in °C).
For example, 40°C. From 40°C, the temperature became 29°C. Then convert it to °F
40°C = 104°F
29°C = 84.2°F
104 - 84.2 = 19.8
Solution #2:
Multiply the change in °C by 1.8
11*1.8 = 19.8
4.) Solve for x in the equation |2x - 1| = 5
Answer: -2 or 3
Solution:
2x - 1 can be 5 or -5 since we are looking for its absolute value
Case 1: 2x - 1 is positive
2x - 1 = 5
x = 3
Case 2: 2x - 1 is negative
-(2x - 1) = 5
-2x + 1 = 5
x = -2
By substituting back the values of x to the original equation, we can conclude that it both satisfy the equation. (NOTE: It is important to substitute back the values of x if it satisfies the equation because there are instances that a value doesn't satisfy the original equation)
5.) What is the slope of the line if its x-intercept is 6 and its y-intercept is 2/3?
Answer: -1/9
Solution:
Slope = - (y-intercept)/(x-intercept)
Slope = -(2/3)/(6)
Slope = -1/9
6.) Write an equation of a line parallel to x - 4y + 8 = 0 and passes through the point (3,-1).
Answer: x - 4y - 7 = 0
Solution:
Parallel lines have the same slopes so just copy x - 4y then equate to C.
x - 4y = C
x - 4y = (3) - 4(-1)
x - 4y = 7
x - 4y - 7 = 0
7.) Write an equation of a line perpendicular to 3x - 7y + 13 = 0 and passes through the point (-7,-1).
Answer: 7x + 3y + 52 = 0
Solution:
Perpendicular lines have negative reciprocal slopes. Interchange coefficients of x and y then change the signs, then also equate to C.
We will have 7x + 3y = C
7x + 3y = 7(-7) + 3(-1)
7x + 3y = -52
7x + 3y + 52 = 0
8.) Factor completely: (x + 3y)(x + 3y - 3) + 2.
Answer: (x + 3y - 1)(x + 3y - 2)
Solution:
Let a = x + 3y
The expression becomes (a)(a-3) + 2
a² - 3a + 2 = (a - 1)(a - 2)
Substitute back a = x + 3y
(x + 3y - 1)(x + 3y - 2)
9.) How many 4-digit numbers are there such that the last digit is non-zero and no digits are repeated?
Answer: 4032
Solution:
We can express the four digits as ABCD
D = Must be non-zero, we have 9 digits to choose from.
A = Must also be non-zero (because it is the thousands digit) and must also be different digit from D (because no digits are repeated). So we have 8 digits to choose from.
B = Must be different from D and A. So we also have 8 digits to choose from.
C = Myst be different from the other three. So we have 7 digits to choose from.
Multiply. (9)(8)(8)(7) = 4032
10.) If r and s are the roots of x² - 4x + 7 = 0, find r² + s².
Answer: 2
Solution:
Sum of roots (r+s) = -b/a = 4
Product of roots (rs) = c/a = 7
r² + s² = (r + s)² - 2rs
r² + s² = 4² - 2(7)
r² + s² = 2
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-shinnichi
