Explanation

To compare the fractions 1/2, 2/3, and 7/8:

Step 1: Find a common denominator

The least common denominator for 2, 3, and 8 is 24.

Step 2: Convert fractions to have the common denominator

1/2 = 12/24

2/3 = 16/24

7/8 = 21/24

Step 3: Compare the fractions

Since 21 > 16 > 12, the order is 7/8 > 2/3 > 1/2.

Explanation

Given sets A = {1, 2, 5, 3} and B = {3, 4}, and assuming the universal set U = {1, 2, 3, 4, 5}:

Step 1: Find B'

B' = U - B

= {1, 2, 3, 4, 5} - {3, 4}

= {1, 2, 5}

Step 2: Find A ∩ B'

A ∩ B' = {1, 2, 5, 3} ∩ {1, 2, 5}

= {1, 2, 5}

Explanation

Given the equation 4x³ - x² - 11x - 6 = 0 and -1 is a root:

Step 1: Factor out the known root

If -1 is a root, (x + 1) is a factor. Perform polynomial division:

4x³ - x² - 11x - 6 = (x + 1)(4x² - 5x - 6)

Step 2: Factor the quadratic equation

4x² - 5x - 6 = (4x + 3)(x - 2)

Step 3: Find the roots

The factors are (x + 1)(4x + 3)(x - 2) = 0

Roots are x = -1, x = -3/4, and x = 2

The other roots are 2 and -3/4.

Explanation

To convert 96 to base 5:

Divide 96 by 5 repeatedly and note the remainders:

96 ÷ 5 = 19 remainder 1

19 ÷ 5 = 3 remainder 4

3 ÷ 5 = 0 remainder 3

Now write remainders in reverse order:

(341)₅

Explanation

To find the distance between two points (x1, y1) and (x2, y2), use the distance formula:

√((x2 - x1)² + (y2 - y1)²)

Given points (p, q) and (p + 4, q + 5):

Step 1: Apply the distance formula

Distance = √((p + 4 - p)² + (q + 5 - q)²)

= √((4)² + (5)²)

= √(16 + 25)

= √41

Explanation

For a quadratic equation in the form of ax² + bx + c = 0, the sum of the roots is given by -b/a.

Given the equation px² - qx + r = 0:

Step 1: Identify a and b

a = p, b = -q

Step 2: Calculate the sum of roots

Sum of roots = -(-q) / p

= q / p

Explanation

To express 27 minutes 30 seconds and 1 hour in ratio:

Step 1: Convert both times to seconds

27 minutes 30 seconds = (27 * 60) + 30

= 1620 + 30

= 1650 seconds

1 hour = 60 minutes = 60 * 60

= 3600 seconds

Step 2: Calculate the ratio

Ratio = 1650 : 3600

To simplify, divide both parts by their greatest common divisor, which is 150:

= (1650 / 150) : (3600 / 150)

= 11 : 24

Explanation

Given x = 3 - √8:

Step 1: Find 1/x

To rationalize the denominator, first simplify √8 to 2√2.

x = 3 - 2√2

1/x = 1 / (3 - 2√2)

Multiply numerator and denominator by the conjugate (3 + 2√2):

1/x = (3 + 2√2) / ((3 - 2√2)(3 + 2√2))

= (3 + 2√2) / (9 - 8)

= 3 + 2√2

Step 2: Calculate x + 1/x

x + 1/x = (3 - 2√2) + (3 + 2√2)

= 6

Explanation

Given the expression a² - 2a + 1 - b²:

Step 1: Factor the perfect square trinomial

a² - 2a + 1 = (a - 1)²

So, the expression becomes (a - 1)² - b²

Step 2: Apply the difference of squares formula

(a - 1)² - b² = (a - 1 - b)(a - 1 + b)

Explanation

Given the area of the square is 16 cm²:

Step 1: Find the side length

Area = side²

16 = side²

side = √16

side = 4 cm

Step 2: Calculate the diagonal

The diagonal of a square can be found using the formula d = side√2.

d = 4√2