Quadratics and Polynomial Functions
(Advanced Math)
(Advanced Math)
What are the solutions to the equation 2x^2 - 8x - 10 = 0?
x = 5 and x = -1 (Factor as 2(x - 5)(x + 1) = 0).
Rewrite in exponent form "the third root of x to the fifth"
x^(5/3)
Simplify the expression (16x^8)^(1/4) for all x > 0.
2x^2
A car travels 180 miles on 6 gallons of gas. How many miles can it travel on 10 gallons at the same fuel efficiency?
300 miles
An item originally priced at $80 is on sale for $64. What is the percent discount?
20%
A quadratic function is defined by f(x) = 3(x - 4)^2 + 7. What is the minimum value of f(x) and at what value of x does it occur?
Minimum value is 7, occurring at x = 4 (Vertex form (h, k) = (4, 7)).
If (a^(2b)) / (a^c) = a^10 for all positive values of a, express c in terms of b.
c = 2b - 10 (Exponent rule: 2b - c = 10).
Solve for x: sqrt(2x + 1) = 5.
x = 12
If 3 printers can print 150 pages in 5 minutes, how many pages can 5 printers print in 10 minutes at the same rate?
500 pages
A set of 5 numbers has a mean of 12. If a 6th number, 24, is added to the set, what is the new mean?
14
The polynomial p(x) = x^3 - 4x^2 + kx - 12 has a factor of (x - 3). What is the value of k?
k = 7 (By Remainder Theorem, p(3) = 0, so 27 - 36 + 3k - 12 = 0, which means 3k = 21).
Find all intersection points (x, y) of the system: y = x^2 - 4 and y = 3x.
(4, 12) and (-1, -3) (Set x^2 - 3x - 4 = 0, so (x - 4)(x + 1) = 0).
A sample of bacteria doubles every 4 hours. If the initial population is 150, write an exponential function P(t) for the population after t hours.
P(t) = 150 * 2^(t/4)
A recipe requires 3/4 cup of sugar for every 2 cups of flour. How many cups of sugar are needed for 7 cups of flour?
2.625 cups (or 21/8)
A price increases by 20%, and then the new price decreases by 20%. What is the overall net percent change from the original price?
4% decrease
For what value of c does the equation 4x^2 - 12x + c = 0 have exactly one real solution?
c = 9 (Set discriminant b^2 - 4ac = 0, so 144 - 16c = 0, which means c = 9).
Which expression is equivalent to (2x^2 + 5x - 3) / (x + 3) for x is not equal to -3?
2x - 1 (Factor numerator: (2x - 1)(x + 3)).
Solve for x: sqrt(x + 6) = x.
x = 3 (Note: x = -2 is extraneous)
A runner completes a race at a constant speed of 12 kilometers per hour. What is the runner's speed in meters per second?
3.33 m/s (or 10/3 m/s)
Dataset A: {10, 11, 12, 13, 14}. Dataset B: {2, 6, 12, 18, 22}. Which dataset has the larger standard deviation?
Dataset B
The graph of y = a(x + 2)(x - 6) passes through the point (1, -45). What is the value of a?
a = 3 (Substitute (1, -45): -45 = a(3)(-5), so -15a = -45, which means a = 3).
If (1 / a) + (1 / b) = 1 / c, solve for a in terms of b and c.
a = (b * c) / (b - c) (Common denominator gives 1 / a = (b - c) / (b * c), then invert).
An investment loses 12% of its value each year. If its initial value is V_0, after how many years will it first be worth less than V_0 * (0.88)^5?
6
On a map scale, 0.5 inches represents 12 miles. On the map, a park has an area of 3 square inches. What is the actual area of the park in square miles?
1,728 square miles
A data set of 20 distinct integers has a median of 45. If the largest value in the set increases by 100, how do the mean and median change?
Mean increases by 5; median stays unchanged.