Logarhythm

The line x = -3 goes through point C on the triangle. This means point C stays the same after the reflection. 

Then just translate it 4 units right and 1 unit up:

(-3 + 4, 0 + 1)

C" = (1, 1) 

Point A after the reflection will be 4 units to the right. This is because the line of reflection is vertical, and point A is 2 units left of the line. 

(-5 + 4, 2) 

A' = (-1, 2)

Then translate it just like you did for point C:

(-1 + 4, 2 + 1)

A" = (3, 3) 

Finally, you can do the same thing in A for point B:

(-6 + 6, -1)

B' = (0, -1)

(0 + 4, -1 + 1)

B" = (4, 0) 

Let me know if you have any questions. 

let x and y represent the two integers respectively. 

we know that their sum is 24. we could then isolate for one variable:

x + y = 24

      y = 24 - x

their product would then be: 

xy = x(24 - x)

     = 24x - x² 

     = -(x² - 24x) 

     = -(x - 24x + 144 - 144) 

     = -[(x - 12)² - 144]

     = -(x - 12)² + 144

In vertex form, we can see the maximum. 

The answer is 144.

The x-intercept will always have a y-coordinate of 0. To find the x-intercepts of the two lines, substitute 0 for y: 

First line: 

   \(y = \frac{1}{3}x + 7\)

\((0) = \frac{1}{3}x + 7\)

\(-7 = \frac{1}{3}x\)

   \(x = -21\)

Second line: 

   \(y = 2x - 5\)

\((0) = 2x - 5\)

   \(5 = 2x\)

   \(x = 2.5\)

Their sum: 

2.5 + (-21) = -18.5

Let me know if you have any questions 

x⁴ - 5x³ + 5x² + 17x - 42 - x⁴ + 5x³ - 4x² - 16x + 20

= x⁴ - x⁴ - 5x³ + 5x³ + 5x² - 4x² + 17x - 16x - 42 + 20

= x² + x - 22

\(x = \frac{-1 + \sqrt{89}}{2}\)

\(x = \frac{-1 - \sqrt{89}}{2} \)

sinθ = opposite/hypotenuse

hence because sinR = 2/5,

opposite = 2 units

hypotenuse = 5 units

using the pythagorean theorem, the adjacent side to angle R would be:

\(=\sqrt{5^2 - 2^2}\)

\(= \sqrt{21}\)

therefore sinT = \(\frac{\sqrt{21}}{5}\)

g⁴ + 18g² + 24 

= g⁴ + 18g² + 81 - 81 + 24

= (g² + 9)² - 57

q = -57

when I use "(i), (ii), (iii)" I'm just labeling the equations

   (i) x + y - z = -8
  (ii) x - y + z = 18
(iii) -x + y + z = 30

   (i) x + y - z = -8

  (ii) x - y + z = 18

      -------------------

                2x = 10

                  x = 5

    (ii) x - y + z = 18

  (iii) -x + y + z = 30

       --------------------

                   2z = 48

                     z = 24

        i) x + y - z = -8

    (5) + y - (24) = -8
               y - 19 = -8

                      y = 11

therefore xyz = 5 * 11 * 24

                      = 1320

when two lines are parallel, their slopes are the same

hence h, in the equation y = hx + k, would be 12

to find k, you'd substitue point (-6, 4) into the existing equation:
y = 12x + k

4 = 12(-6) + k

4 = -72 + k

k = 76

so the value of k is 76

(the equation of the line would thus be y = 12x + 76)

a Mersenne prime is a prime number one less than a power of two

2⁷ = 128 which means there are 6 number that can be a Mersenne prime less than 100: 

2¹ - 1 = 1, which isn't prime
2² - 1 = 3, which is prime

2³ - 1 = 7, which is prime
2⁴ - 1 = 15, which isnt prime

2⁵ - 1 = 31, which is prime

2⁶ - 1 = 63, which isnt prime 

this means there are 3 Mersenne primes that are less than 100

-5p + 20 - 5p + 20

= -5(p - 4 + p - 4)

= -5(2p - 8)

I understand it now. The question was just poorly written. 

              \(b^2 - 4ac = 7\)

\(b^2 - 4(1)(b + 3) = 7\) 
       \(b^2 - 4b - 12 = 7\)
       \(b^2 - 4b - 19 = 0\) 
                            \(b = {-(-4) \pm \sqrt{(-4)^2-4(1)(-19)} \over 2(1)}\) 

                            \(b = {4 \pm \sqrt{16+ 76} \over 2}\)

                            \(b = {4 \pm 2\sqrt{23} \over 2}\)

                            \(b = 2\pm\sqrt{23}\)

                            \(b>0 \)

                        \(\therefore b = 2 + \sqrt{23}\)

                                      \(25x^2 + 40x - 10 = 0\)
\(25x^2 + 40x + (\frac{40\div2}{5})^2 - (\frac{40\div2}{5})^2 - 10 = 0\)

                   \(25x^2 + 40x + 16 - 16 - 10 = 0\)

                                \((5x + 4)^2 - 16 - 10 = 0 \)

                                                    \((5x + 4)^2 = 26\)

therefore the value of a + b + c is 5 + 4 + 26, which equals 35. 

the value of the 7 in 735208 is "700000"
the value of the 7 in 137342 is "7000"

7000 x 10 ≠ 700000

therefore this isn't correct. 

you can approach this question multiple ways. 
start by finding the equation of this graph. you can do this in vertex or factored form for ease:

y = a(x + 1)² - 1    or    y = a(x)(x + 2)

now pick a point and substitute it in to find the dilation factor, a. 

(1, 3)  
3 = a(1 + 1)² - 1    or    3 = a(1)(1 + 2)
3 = 4a - 1                     3 = 3a
4 = 4a                          a = 1
a = 1

y = 1(x + 1)² - 1            y = 1x(x + 2)
y = (x + 1)² - 1              y = x(x + 2)

now you just convert the equation into standard form by expanding: 

y = (x + 1)² - 1       or    y = x(x + 2)
y = x² + 2x + 1 - 1         y = x² + 2x
y = x² + 2x

therefore a + b + c = 1 + 2 + 0
                               = 3. 

as said, start by isolating y: 
5x + y = 31
        y = 31 - 5x

then substituting that into the second equation: 
3x + 4y = -37
3x + 4(31 - 5x) = -37
3x + 124 - 20x = -37
        124 - 17x = -37
               - 17x = -161
                     x = \(9\frac{8}{17}\)

now you can substitute the x value back into the first equation to find y: 
y = 31 - 5x 
   = 31 - 5( \(9\frac{8}{17}\) )
   = 31 - \(47\frac{6}{17}\)
   = \(-16\frac{6}{17}\)

so your answer would be \((9\frac{8}{17}, -16\frac{6}{17})\). 

hope this helps and tell me if I did anything wrong :) 

  (2x + 7)(x - 5) = -43 + jx + x²
    2x² - 3x - 35 = -43 + jx + x²
 x² - 3x - jx + 8 = 0
x² - (3 + j)x + 8 = 0

when there is only one real solution, by the quadratic equation:
                b² - 4ac = 0
(-(3 + j))² - 4(1)(8) = 0
              (-(3 + j))² = 32
                  -(3 + j) = \(±\sqrt{32}\)
                     3 + j = \(±4\sqrt{2}\)
                           j = \(±4\sqrt{2} - 3\)

\(4\sqrt{2} - 3 ≈ 2.66\)
\(-4\sqrt{2} - 3 ≈ -8.66\)

which then your answer is -8.66, 2.66

please tell me if I did anything wrong. hope this helps

f(x) = x² - 2x + 5

g(x) = x + 8

f(g(5)) - g(f(5))
= [(g(5))² - 2(g(5)) + 5] - [(f(5)) + 8]
= [((5) + 8)² - 2((5)) + 5] - [((5)² - 2(5) + 5) + 8]
= [169 - 10 + 5] - [25 - 10 + 5 + 8]
= 164 - 28
= 136

 

we know the point of intersection is (-4, -7), so k has to be 0 since the y intercept of the equation y = kx - 7 is (0, -7), it follows that the line is horizontal therefore it has a slope of 0.