Explanation

Equation: (4x + 1)/5 = (x + 9)/2

Step 1: Cross multiply to remove fractions

2(4x + 1) = 5(x + 9)

Step 2: Expand both sides

8x + 2 = 5x + 45

Step 3: Move x terms to one side, numbers to the other side

8x - 5x = 45 - 2

3x = 43

Step 4: Divide by 3

x = 43/3

Last verified on 25-05-2026

Explanation

Original equation: x^2 - 11x + 30 = 0

Factor it to find roots:

x^2 - 11x + 30 = (x - 5)(x - 6) = 0

So roots are 5 and 6.

Increase each root by 2:

New roots are 5 + 2 = 7 and 6 + 2 = 8.

New equation with roots 7 and 8 is:

(x - 7)(x - 8) = 0

x^2 - 8x - 7x + 56 = 0

x^2 - 15x + 56 = 0

Last verified on 25-05-2026

Explanation

Let the factors be (x - n) and (x - n - 1).

Multiply: (x - n)(x - n - 1) = x^2 - (2n + 1)x + n(n + 1).

Set n(n + 1) = 110, which gives n = 10.

Then p = 2n + 1 = 21.

Last verified on 25-05-2026

Explanation

If you multiply (a + bi) and (a - bi), you get:

(a + bi)(a - bi) = a_a - a_bi + bi_a - bi_bi

                 = a^2 - abi + abi - b^2_i^2

The middle terms cancel out. And i^2 = -1, so -b^2_i^2 = -b^2*(-1) = b^2.

That leaves a^2 + b^2.

You are told (a + bi)(a - bi) = 89, so a^2 + b^2 = 89.

Last verified on 25-05-2026

Explanation

To get a:c, multiply the two ratios together so the b cancels out.

a:c = (a/b) × (b/c) = 12/13 × 8/9 = 96/117.

96 and 117 can both be divided by 3, which gives 32/39.

So a:c = 32:39.

Last verified on 25-05-2026

Explanation

Add them up and divide by how many numbers there are.

16 + 18 + 20 = 54  

54 ÷ 3 = 18

So the mean is 18.

Last verified on 25-05-2026

Explanation

A secant is a straight line that intersects a circle.

It passes through the circle and cuts it at two points.

Last verified on 25-05-2026

Explanation

cosecθ is just 1 over sinθ.  

So sinθ × cosecθ = sinθ × (1/sinθ) = 1.

The sinθ cancels out, and you’re left with 1.

Last verified on 25-05-2026

Explanation

If sinθ = 9/41, use the identity sin²θ + cos²θ = 1.

First find sin²θ: (9/41)² = 81/1681.  

Then cos²θ = 1 - 81/1681 = 1600/1681.  

Take the square root and you get cosθ = 40/41.

Last verified on 25-05-2026

Explanation

If a/b = 11/12 and b/c = 7/8, then to get a/c you can multiply them together.

a/c = (a/b) × (b/c) = 11/12 × 7/8  

When you multiply, the b cancels out:

a/c = 77/96  

So a:c = 77:96.

Last verified on 25-05-2026