Explanation

Given:

P(x) = x^2 - 4x + 3

One root is 1. Let's find the other root.

The sum of roots = -(-4) = 4 (from the coefficient of x)

One root is 1, so the other root would be:

4 - 1 = 3

Explanation

Given:

a + b = 4

ab = 3

The formula for a^3 + b^3 is:

a^3 + b^3 = (a + b)(a^2 - ab + b^2)

However, a simpler approach uses the identity:

a^3 + b^3 = (a + b)^3 - 3ab(a + b)

Plugging in the values:

a^3 + b^3 = (4)^3 - 3(3)(4)

= 64 - 36

= 28

Explanation

The calculation confirms that:

3i(2 + 5i) = 6i + 15i^2

= 6i - 15

Comparing this to x + 6i:

x = -15

Explanation

The discriminant (D) of a quadratic equation ax^2 + bx + c = 0 is given by:

D = b^2 - 4ac

Given:

a = 1, b = 9, c = 2

D = 9^2 - 4(1)(2)

= 81 - 8

= 73

Explanation

Given:

(x + 1)(x - 5) = 0

This implies:

x + 1 = 0 or x - 5 = 0

Solving for x:

x = -1 or x = 5

Explanation

The equation is:

(y + 13)(y + a) = 0

Expanding it:

y^2 + ay + 13y + 13a = 0

y^2 + (a + 13)y + 13a = 0

For no linear term, the coefficient of y should be 0:

a + 13 = 0

a = -13

Explanation

The traveler covers:

First journey: x km

Second journey: y km

Third journey: x km

Total distance = x + y + x = 2x + y

Explanation

Given:

Sum of numbers = -8

Product of numbers = 15

The algebraic expression representing this is:

x^2 - (sum)x + (product) = 0

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

x^2 + 8x + 15 = 0

The correct answer is: x^2 + 8x + 15

Explanation

When you multiply two numbers with different signs (one positive and one negative), the result is always negative. In this case, (-2) multiplied by (+3) equals (-6).

Here's the calculation:

(-2) x (+3) = -2 x 3 = -6

So, the correct answer is -6.

Explanation

Given:

x + 1/x = 10

Squaring both sides:

(x + 1/x)^2 = 10^2

x^2 + 2(x)(1/x) + 1/x^2 = 100

x^2 + 2 + 1/x^2 = 100

Subtract 2 from both sides:

x^2 + 1/x^2 = 98