Explanation

Given:

Z1 = 2 - i

Z2 = 3 + i

Product of Z1 and Z2:

Z1 * Z2 = (2 - i) * (3 + i)

= 6 + 2i - 3i - i²

Since i² = -1,

= 6 - i + 1

= 7 - i

Explanation

To divide (3x³ - 19x² + 10x + 50) by (x - 5), let's perform polynomial division:

1. Divide 3x³ by x: 3x²

2. Multiply (x - 5) by 3x²: 3x³ - 15x²

3. Subtract: -19x² + 15x² = -4x², bring down 10x

4. Divide -4x² by x: -4x

5. Multiply (x - 5) by -4x: -4x² + 20x

6. Subtract: 10x - 20x = -10x, bring down 50

7. Divide -10x by x: -10

8. Multiply (x - 5) by -10: -10x + 50

9. Subtract: 0

The quotient is:

3x² - 4x - 10

Explanation

Given the equations:

1. 7x - 4y = 10

2. 2x + 3y = 65

Let's solve them using elimination or substitution. Multiplying equation 1 by 3 and equation 2 by 4 gives us a way to eliminate y:

1. 21x - 12y = 30

2. 8x + 12y = 260

Adding both equations:

29x = 290

x = 290 / 29

x = 10

Explanation

Given the sequence formula:

an = 3n + 5

To find the 7th term, substitute n = 7:

a7 = 3(7) + 5

= 21 + 5

= 26

Explanation

To find (-4a^2)^3:

(-4a^2)^3 = (-4)^3 * (a^2)^3

= -64 * a^(2*3)

= -64 * a^6

= -64a^6

Explanation

To divide (2x^3 - 5x^2y + 5xy^2 - 3y^3) by (2x - 3y), let's attempt polynomial long division or factorization if possible.

Given the form of the dividend and divisor, let's try to factor or directly apply division:

(2x^3 - 5x^2y + 5xy^2 - 3y^3) ÷ (2x - 3y)

Upon inspection or performing polynomial division, one notices:

2x^3 - 5x^2y + 5xy^2 - 3y^3 can be factored or divided as follows:

2x^3 - 3x^2y - 2x^2y + 3xy^2 + 2xy^2 - 3y^3

Which can be rearranged and factored:

x^2(2x - 3y) - xy(2x - 3y) + y^2(2x - 3y)

= (2x - 3y)(x^2 - xy + y^2)

Thus, (2x^3 - 5x^2y + 5xy^2 - 3y^3) ÷ (2x - 3y) = x^2 - xy + y^2

Explanation

"6 times a number" = 6x.

"Less than" means subtraction: 6x - 79 = x represents the statement.

Explanation

Given:

Arc length = 2.5 cm

Diameter = 6 cm, so radius (r) = 3 cm

The formula for arc length is:

Arc length = θr

2.5 = θ * 3

θ = 2.5 / 3

θ = 0.833

Explanation

Since y varies jointly as x and z:

y = kxz

Given y = 33 when x = 9 and z = 12:

33 = k * 9 * 12

33 = 108k

k = 33/108

k = 11/36

Now, find y when x = 16 and z = 22:

y = (11/36) * 16 * 22

y = (11 * 16 * 22) / 36

y = 3872 / 36

y = 968 / 9

Explanation

Given the quadratic equation:

mx² - 5x + n = 0

Sum of roots = -(-5)/m = 5/m

Product of roots = n/m

Given:

Sum of roots = Product of roots = 10

So, 5/m = 10 => m = 1/2

And n/m = 10 => n = 10m = 10 * (1/2) = 5