Explanation

Given x + 1/x = 10:

Step 1: Square both sides

(x + 1/x)² = 10²

x² + 2(x)(1/x) + 1/x² = 100

x² + 2 + 1/x² = 100

Step 2: Solve for x² + 1/x²

x² + 1/x² = 100 - 2

x² + 1/x² = 98

Explanation

f(x) = x², g(x) = 2x, so

fog(x) = f(g(x)) = f(2x) = (2x)² = 4x²

Explanation

Given the equation 6x + 5(3 - 1.4x) = 1.7:

Step 1: Distribute and simplify

6x + 15 - 7x = 1.7

-x + 15 = 1.7

Step 2: Solve for x

-x = 1.7 - 15

-x = -13.3

x = 13.3

Explanation

Given cosx(tan x + cot x):

Step 1: Express tan x and cot x in terms of sin x and cos x

tan x = sin x / cos x

cot x = cos x / sin x

Step 2: Substitute and simplify

cosx(tan x + cot x) = cosx(sin x / cos x + cos x / sin x)

= sin x + cos² x / sin x

= (sin² x + cos² x) / sin x

= 1 / sin x (since sin² x + cos² x = 1)

Step 3: Identify the result

1 / sin x = cosec x

Explanation

Given the quadratic equation x² + 2x - 2 = 0:

Step 1: Apply the quadratic formula

x = (-b ± √(b² - 4ac)) / 2a

Here, a = 1, b = 2, and c = -2.

Step 2: Calculate the roots

x = (-(2) ± √((2)² - 4_1_(-2))) / 2*1

x = (-2 ± √(4 + 8)) / 2

x = (-2 ± √12) / 2

x = (-2 ± 2√3) / 2

x = -1 ± √3

Explanation

Given the equation 4x³ - x² - 11x - 6 = 0 and -1 is a root:

Step 1: Factor out the known root

If -1 is a root, (x + 1) is a factor. Perform polynomial division:

4x³ - x² - 11x - 6 = (x + 1)(4x² - 5x - 6)

Step 2: Factor the quadratic equation

4x² - 5x - 6 = (4x + 3)(x - 2)

Step 3: Find the roots

The factors are (x + 1)(4x + 3)(x - 2) = 0

Roots are x = -1, x = -3/4, and x = 2

The other roots are 2 and -3/4.

Explanation

To find the distance between two points (x1, y1) and (x2, y2), use the distance formula:

√((x2 - x1)² + (y2 - y1)²)

Given points (p, q) and (p + 4, q + 5):

Step 1: Apply the distance formula

Distance = √((p + 4 - p)² + (q + 5 - q)²)

= √((4)² + (5)²)

= √(16 + 25)

= √41

Explanation

For a quadratic equation in the form of ax² + bx + c = 0, the sum of the roots is given by -b/a.

Given the equation px² - qx + r = 0:

Step 1: Identify a and b

a = p, b = -q

Step 2: Calculate the sum of roots

Sum of roots = -(-q) / p

= q / p

Explanation

Given x = 3 - √8:

Step 1: Find 1/x

To rationalize the denominator, first simplify √8 to 2√2.

x = 3 - 2√2

1/x = 1 / (3 - 2√2)

Multiply numerator and denominator by the conjugate (3 + 2√2):

1/x = (3 + 2√2) / ((3 - 2√2)(3 + 2√2))

= (3 + 2√2) / (9 - 8)

= 3 + 2√2

Step 2: Calculate x + 1/x

x + 1/x = (3 - 2√2) + (3 + 2√2)

= 6

Explanation

Given the expression a² - 2a + 1 - b²:

Step 1: Factor the perfect square trinomial

a² - 2a + 1 = (a - 1)²

So, the expression becomes (a - 1)² - b²

Step 2: Apply the difference of squares formula

(a - 1)² - b² = (a - 1 - b)(a - 1 + b)