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## Nataliya D.'s Resources

x + y = 3 +     x - y = 7      2x = 10 ----> x = 5   5 + y = 3 ----> y = - 2 5 + (- 2) = 3  5 - (- 2) = 7

- 2 ± 2 i √3   ---------------- =         2    2( - 1 ± i √3)  ------------------- =            2    - 1 ± i √3

Rule for Δ ABC: any two sides added together must be greater than the third side. If       AB = c       BC = a       AC = b, then     a + b > c     a +...

√(5x - 1) = 1 + √(x + 2) √(5x - 1) - √(x + 2) = 1 [√(5x - 1) - √(x + 2)]2 = 12 [√(5x - 1)]2 - 2√(5x - 1) * √(x + 2) + [√(x + 2)]2 = 1  5x - 1 - 2√[(5x - 1)(x + 2)] + x + 2 = 1 - 2√[(5x - 1)(x + 2)]...

(1)  ...  x : y = 5 : 2  (2)  ...  y : z = 8 : 11 Let's rewrite (1) as            x : y = 20 : 8 , and            y : z = 8 : 11 Now,...

√(5x - 1) = 1 [√(5x - 1)]2 = 12 5x - 1 = 1  5x = 2 x = 2/5   Let's check solution in the original equation √(5 * 2/5 - 1) = 1 √(2 - 1) = 1 √1 = 1   1 = 1 Thus, x = 2/5 is not an extraneous root.

x2 - 4y2 + 7 = 0  ..... (1)  3x2 + y2 = 31      ..... (2)  Let's multiply (2) by 4 and add two equations:       x2 - 4y2 = - 7   +   12x2 + 4y2 = 124    ...

sin2 x + cos2 x = 1               1 csc x = ------             sin x               1 sec...

In the set of real numbers for any a ≥ 0 and b ≥ 0 : √(ab) = √a * √b  ~~~~~~~~~~~~~~  √(6a)√(3a) + √(2a) (a ≥ 0) √(18a2) + √2 * √a = √(9a2) * √2 + √2 * √a =  3a√2 + √2 * √a =  √2(3a + √a...

(C) a = 0 produces a counter example. If a = 0, then we have f(x) = 0x + b or f(x) = b , which is a horizontal line. Thus, the range (all possible values of "y") is limited to whatever "b" is.

3x2 - 2x + 1 = 5  3x2 - 2x - 4 = 0 . Let's multiply this equation by 3 9x2 - 6x - 12 = 0 9x2 - 6x = 12 (3x)2 - 2*3x*1 + 1 = 12 + 1 (3x - 1)2 = 13 3x - 1 = ±√13  3x = 1 ± √13        ...

(a ± b)2 = a2 ± 2ab + b2   (f o g)(x) = f(g(x)) ~~~~~~~~~~~~~~  f(x) = 8x2 - 7x g(x) = 8x - 4 (f o g)(x) = f[g(x)] = 8(8x - 4)2 - 7(8x - 4) = 8(64x2 - 64x + 16) - 56x + 28 = 512x2 - 512x + 128 - 56x +...

logcan = n*logca                                 a logca - logcb = logc ---                                 ...

Vertical line has undefined slope. Thus, the equation will be x = - 3 , or for any value of "y" value of "x" is always "- 3"

100% = 1 A = length * width "... if each of the dimensions of a rectangle is increased by 100% ..." Let length of new rectangle be "l1" and width - "w1" l1 = l + l * 100% = l + l * 1 = 2l w1 = w + w * 100% = w...

Rule for Δ ABC: any two sides added together must be greater than the third side. If    AB = c    BC = a    AC = b, then    a + b > c    a + c > b    b + c > a ~~~~~~~~~~~~...

Angles ACB and DGC are on opposite sides of transversal line AC and two parallel lines DF and HI. Thus, angle ACB and DGC are alternate interior angles, therefore measure of angel ACB = measure of angel DGC  m < ACB = m < DGC = 58.6o < CGD and < AGD are supplementary...