Algebraic Equations and Expressions

The continued fraction x = 1/(6+1/(2+1/(6+1/(2+...)))) can be solved by noting the periodic pattern. Let x = 1/(6+1/(2+x)), which leads to a quadratic equation. Solving gives x = (-3+2√3)/3, which matches option A in decimal form.

The expression (a-x)(b-x)...(z-x) contains the term (x-x) because the sequence includes every letter of the alphabet subtracted by x. Since (x-x) = 0, the entire product becomes 0 regardless of the value of other terms.

Given i^2 = -1, we calculate: i^3 = i × i^2 = i × (-1) = -i. Numerator: i + i^2 + i^3 = i + (-1) + (-i) = -1. The fraction becomes (-1)/(-i) = 1/i. Multiplying numerator and denominator by i: i/(i^2) = i/(-1) = -i.

To find x, divide 1050 by 1400: x = 1050/1400 = 3/4. This simplifies to 0.75. Option D (3/4) is correct. The other options (1/4, 2/3, 3/5) do not equal 0.75 when calculated.

If x^4 = 16, then x = 2 (since 2*2*2*2 = 16). Substituting x = 2 into the second expression, 4^x becomes 4^2, which equals 16.

For the first option, if $x=2$, then $3x(7x+12) = 3(2)(7(2)+12) = 6(14+12) = 6(26) = 156$, not 94. Thus, this equation is incorrect, making it the correct choice.

Combine like terms: 15x + x = 16x. The equation becomes 16x = 80. Dividing both sides by 16 gives x = 5.

The subs function syntax is subs(expression, variable, value). For f=3*x^2, subs(f, x, 5) substitutes x=5. Options A has wrong syntax with double comma. Options B and C swap arguments - subs expects (f, variable, value), not (f, value, variable) or (value, f, variable).

The expression follows standard operator precedence: multiplication and division are evaluated left to right first (10*2=20, 20/5=4, 4*2=8), then addition (8+25=33). Option A (31) and B (32) are incorrect calculations. Option D (34) would be the result if the final addition used 26 instead of 25.

{- 4/3 x 5/4 x - 3/5 x 6/7 x (14/12 x 13/11 x 33/39) } / 2

= {- 4/3 x 5/4 x - 3/5 x 6/7 x 14/12 x 13/11 x 33/39} / 2

= {1 x 6/7 x 7/6 x 13/11 x 11/13} / 2

= {1}/2

= 1/2

6x - 40 = -16 Adding 40 to both sides 6x = -16 + 40 6x = 24 Dividing both sides by 6 x = 4

8x + 8 = 4x Subject term on one side and non subject term on other side. 8x - 4x = -8 4x = -8 Dividing both sides by 4 x = -2

6x + 18 = 4x Subject terms on one side and on subject terms on other side. 6x - 4x = -18 2x = -18 Dividing both sides by 2 x = -9  

-15x - 5 = -4x - 16 Bring subject terms on one side and non subject terms to other side. 15x - 4x = 16 - 5 11x = 11 Dividing both sides by 11 x = 1

2x + 20 = 24x + 42 Bring subject terms on one side and non subject terms to another side. 24x - 2x = -42 + 20 22x = -22 Dividing both sides by 22 x = -1