Explanation

For f(x) = x² - x, find f(5) and f(3) separately then subtract.

f(5) = 5² - 5 = 25 - 5 = 20  

f(3) = 3² - 3 = 9 - 3 = 6  

Now f(5) - f(3) = 20 - 6 = 14  

Last verified on 25-05-2026

Explanation

y ∝ 1/x³ means y = k / x³ where k is a constant.

Step 1: Find k using y = 32 when x = 2

32 = k / 2³

32 = k / 8

k = 32 * 8 = 256

Step 2: Use k to find y when x = 4

y = 256 / 4³

y = 256 / 64

y = 4

Last verified on 25-05-2026

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