Explanation

x - 8/3 + x - 3/2 = 0

Multiply both sides by 6:

2(x - 8) + 3(x - 3) = 0

2x - 16 + 3x - 9 = 0

5x = 25

x = 5

Explanation

Given x = 1 is a solution, let's find 'a':

a(1)² + 5(1) - 3 = 0

a + 2 = 0

a = -2

The equation is -2x² + 5x - 3 = 0.

Let's factor or solve:

-2x² + 5x - 3 = 0

-2x² + 2x + 3x - 3 = 0

-2x(x - 1) + 3(x - 1) = 0

(-2x + 3)(x - 1) = 0

x = 1 (given) or x = 3/2

The other solution is x = 3/2.

Explanation

Given:

1. Sum of roots: α + β = 2

2. Sum of cubes of roots: α³ + β³ = 98

Step-by-Step Solution

Step 1: Use the identity for α³ + β³

α³ + β³ = (α + β)³ - 3αβ(α + β)

Step 2: Substitute known values

98 = (2)³ - 3αβ(2)

98 = 8 - 6αβ

Step 3: Solve for αβ

98 - 8 = -6αβ

90 = -6αβ

αβ = -15

Step 4: Form the quadratic equation

x² - (α + β)x + αβ = 0

x² - 2x - 15 = 0

The answer is x² - 2x - 15 = 0.

Explanation

tan θ = sin θ/cos θ and cot θ = cos θ/sin θ

tan θ + cot θ = sin θ/cos θ + cot θ = cos θ/sin θ

= sin^2 θ + cos^2 θ/sin θ cos θ

= 1/sin θ cos θ (since sin² θ + cos² θ = 1)

Now, 1/sin θ cos θ = 1/sin θ × 1/cos θ = cosec θ × sec θ = sec θ cosec θ

The answer is sec θ cosec θ.

Explanation

The line y = 3 is a horizontal line that intersects the y-axis at y = 3.

Since it's horizontal, it's parallel to the x-axis.