Explanation

If a = 10, then put it in the equation:

a² + b² = 200  

10² + b² = 200  

100 + b² = 200  

Subtract 100 from both sides:

b² = 200 - 100  

b² = 100  

So b = 10.

Last verified on 25-05-2026

Explanation

(12x + 8)/(x(x + 4)) = A/x + B/(x + 4)

Multiply both sides by x(x + 4):

12x + 8 = A(x + 4) + B(x)

Put x = 0:

12(0) + 8 = A(0 + 4) + B(0)

8 = 4A

A = 2

Put x = -4:

12(-4) + 8 = A(-4 + 4) + B(-4)

-48 + 8 = -4B

-40 = -4B

B = 10

Therefore:

(12x + 8)/(x(x + 4)) = 2/x + 10/(x + 4)

Last verified on 25-05-2026

Explanation

f(x) = x² - 4x

f(4) = 4² - 4(4) = 16 - 16 = 0  

f(2) = 2² - 4(2) = 4 - 8 = -4  

f(4) - f(2) = 0 - (-4) = 4

Last verified on 25-05-2026

Explanation

Since y ∝ x³, we have y = kx³ for some constant k.

1. Find k using y = 54 when x = 3:  

54 = k(3³) = k(27)  

k = 54 / 27 = 2

2. Now find y when x = 2:  

y = 2(2³) = 2(8) = 16

Last verified on 25-05-2026

Explanation

(3x + 2)/4 = (x + 6)/2

Multiply both sides by 4:

3x + 2 = 2(x + 6)

Expand RHS:

3x + 2 = 2x + 12

Collect x terms:

3x - 2x = 12 - 2

x = 10

Last verified on 25-05-2026

Explanation

Original: x² - 10x + 21 = 0  

Substitute x → x + 1:

(x + 1)² - 10(x + 1) + 21 = 0  

x² + 2x + 1 - 10x - 10 + 21 = 0  

x² - 8x + 12 = 0

Last verified on 25-05-2026

Explanation

Let the factors be (x - m) and (x - n)

Then:

m + n = p

m * n = 90

|m - n| = 9

Using (m - n)^2 = (m + n)^2 - 4mn:

9^2 = p^2 - 4*90

81 = p^2 - 360

p^2 = 441

p = ±21

Last verified on 25-05-2026

Explanation

(a + bi)(a - bi) = a² - (bi)² = a² - b²i² = a² - b²(-1) = a² + b²

So (a + bi)(a - bi) = a² + b² = 73

Last verified on 25-05-2026

Explanation

Put a = 9 in a² + b² = 145

9² + b² = 145

81 + b² = 145

b² = 145 - 81

b² = 64

b = ±8

Last verified on 24-05-2026

Explanation

(10x + 6)/(x(x + 3)) decomposes as 2/x + 8/(x + 3)

Solution:

Let (10x + 6)/(x(x + 3)) = A/x + B/(x + 3)

Multiply both sides by x(x + 3)

10x + 6 = A(x + 3) + B(x)

Put x = 0

10(0) + 6 = A(0 + 3) + B(0)

6 = 3A

A = 2

Put x = -3

10(-3) + 6 = A(-3 + 3) + B(-3)

-30 + 6 = 0 - 3B

-24 = -3B

B = 8

So the decomposition is:

2/x + 8/(x + 3)

Last verified on 24-05-2026