Explanation

f(x) = x² - 3

Find f(4):  

f(4) = 4² - 3 = 16 - 3 = 13

Find f(2):  

f(2) = 2² - 3 = 4 - 3 = 1

Now f(4) - f(2) = 13 - 1 = 12

Last verified on 23-05-2026

Explanation

y ∝ x^2 means y = k x^2, where k is a constant.

Given: y = 50 when x = 5

Put values:

50 = k _ 5^2

50 = k _ 25

k = 50 / 25

k = 2

Now find y when x = 2:

y = k x^2

y = 2 _ 2^2

y = 2 _ 4

y = 8

Last verified on 23-05-2026

Explanation

5x + 2 = 3x + 10

Subtract 3x from both sides:

2x + 2 = 10

Subtract 2 from both sides:

2x = 8

Divide by 2:

x = 4

Last verified on 23-05-2026

Explanation

Original equation: x^2 - 8x + 15 = 0

Factor it:

x^2 - 8x + 15 = (x - 3)(x - 5) = 0

So roots are 3 and 5.

Increase each root by 2:

New roots = 3 + 2 = 5, 5 + 2 = 7

Form new equation with roots 5 and 7:

(x - 5)(x - 7) = 0

x^2 - 7x - 5x + 35 = 0

x^2 - 12x + 35 = 0

Last verified on 23-05-2026

Explanation

Let the factors be (x - a) and (x - (a+1)) because they differ by 1.

Multiply them:

(x - a)(x - a - 1) = x^2 - (2a + 1)x + a(a + 1)

Compare with x^2 - px + 56:

a(a + 1) = 56

p = 2a + 1

Solve a(a + 1) = 56

a^2 + a - 56 = 0

(a + 8)(a - 7) = 0

So a = 7 or a = -8

If a = 7, then p = 2_7 + 1 = 15

If a = -8, then p = 2_(-8) + 1 = -15

Last verified on 23-05-2026

Explanation

(a + bi)(a - bi) = a^2 - (bi)^2 = a^2 - b^2 _ i^2

Since i^2 = -1, this becomes:

a^2 - b^2 _ (-1) = a^2 + b^2

Given (a + bi)(a - bi) = 53,

a^2 + b^2 = 53

Last verified on 23-05-2025

Explanation

Using trigonometric identity:

tanθ = sinθ/cosθ

So, sinθ divided by cosθ is equal to tanθ.

Last verified on 15-05-2026

Explanation

Given: sinθ = 8/17

cosθ = √(1 − (8/17)²)

= √(1 − 64/289)

= √(225/289)

= 15/17

Last verified on 15-05-2026

Explanation

Given:

a² + b² = 65

a = 1

Put value of a:

1² + b² = 65

1 + b² = 65

Subtract 1 from both sides:

b² = 64

Take square root:

b = 8 (positive value)

So, b = 8.

Last verified on 15-05-2026

Explanation

The decomposition is:

(8x + 6)/[x(x + 2)] = A/x + B/(x + 2)

Multiply both sides by x(x + 2) so the denominators cancel out:

8x + 6 = A(x + 2) + Bx

Now find A and B:

Put x = 0. That makes the Bx term zero, so 6 = 2A. That gives A = 3.  

Put x = -2. That makes the A(x + 2) term zero, so -10 = -2B. That gives B = 5.

So it equals 3/x + 5/(x + 2).

Last verified on 15-05-2026