Explanation

Use identity: tan θ + sec θ = (sin θ/cos θ) + (1/cos θ) = (sin θ + 1)/cos θ

Therefore, tan θ + sec θ = (1 + sin θ)/cos θ.

Explanation

Let's use the quadratic formula to find the roots:

x = (-b ± √(b² - 4ac)) / 2a

For the given equation x² - 3x + k + 1 = 0:

a = 1, b = -3, c = k + 1

The roots differ by unity, so:

√(b² - 4ac) = 1

(-3)² - 4(1)(k + 1) = 1

9 - 4k - 4 = 1

5 - 4k = 1

-4k = -4

k = 1

Explanation

Given:

ab + bc + ca = -4

a² + b² + c² = 17

Using the identity (a + b + c)² = a² + b² + c² + 2(ab + bc + ca):

(a + b + c)² = 17 + 2(-4)

= 17 - 8

= 9

a + b + c = ±√9

= ±3

Explanation

We are given:

log₅ 125 = x

We know that:

125 = 5³

So,

log₅ 125 = log₅ (5³) = 3

Explanation

Let's analyze the sequence:

-5, 7, 19, 31

The differences between consecutive terms are:

7 - (-5) = 12

19 - 7 = 12

31 - 19 = 12

The sequence has a common difference of 12.

The general term for an arithmetic sequence is:

aₙ = a₁ + (n - 1)d

where a₁ is the first term (-5) and d is the common difference (12).

aₙ = -5 + (n - 1)12

aₙ = -5 + 12(n - 1)

Explanation

To factorize 4x² + 7x + 3:

Let's find two numbers that multiply to 4*3 = 12 and add to 7:

The numbers are 4 and 3 (since 4 + 3 = 7 and 4 * 3 = 12).

Now, rewrite the middle term:

4x² + 4x + 3x + 3

Factor by grouping:

4x(x + 1) + 3(x + 1)

(4x + 3)(x + 1)

The correct answer is (x + 1)(4x + 3).

Explanation

Subtracting a negative number is the same as adding its positive:

(-16) - (-4) = -16 + 4

-16 + 4 = -12

Explanation

Given x < 0 and y < z:

When multiplying both sides of y < z by x, the inequality sign flips because x is negative:

xy > xz

Explanation

Given xy = 1 and x > 0:

y = 1/x

We want to minimize x + y = x + 1/x.

Let's find the minimum value:

Let f(x) = x + 1/x

f'(x) = 1 - 1/x² = 0 for minimum

x² = 1

x = 1 (since x > 0)

f(1) = 1 + 1/1 = 2

The minimum value of x + y is 2.

Explanation

Let's check each pair in the equation:

2x + 3y = 24

{6, 4} → 2(6) + 3(4) = 12 + 12 = 24 ✓

{12, 0} → 2(12) + 3(0) = 24 + 0 = 24 ✓

{0, 8} → 2(0) + 3(8) = 0 + 24 = 24 ✓

{9, -2} → 2(9) + 3(-2) = 18 - 6 = 12 ✗