Algebra discussion

Correct solution.

The answer is x=16.4, y=6.8

Yes!

Solution:

x² - 6x + 9 - 16 = 4y

x² - 6x + 9 = 4y - 16

(x-3)² = 4(y-4)

So, the vertex of the parabola is (3,4) away from the origin √(3²+4²) = 5

Problem 16:

If 4ˣ = 100, find the value of 8ˣ.

Yes.

n = log₄100

Let √(100+√n) = a+√b ; a ∈ Z, b ∈ Z₀+

(a+√b)² = (a²+b) + 2a√b = (a²+b) + √4a²b = 100+√n

(a-√b)² = (a²+b) - 2a√b = (a²+b) - √4a²b = 100-√n

Then √(100+√n)+√(100-√n) = (a+√b)+(a-√b) = 2a is always positive number.

So a²+b = 100...[1] and n = 4a²b...[2]

From equation [1] There are pairs orders (a,b) that can all be as follows.

• (1,99) ; n = 4(1)²(99) = 396
• (2,96) ; n = 4(2)²(96) = 1,536
• (3,91) ; n = 4(3)²(91) = 3,276
• (4,84) ; n = 4(4)²(84) = 5,376
• (5,75) ; n = 4(5)²(75) = 7,500
• (6,64) ; n = 4(6)²(64) = 9,216
• (7,51) ; n = 4(7)²(51) = 9,996
• (8,64) ; n = 4(8)²(36) = 9,216
• (9,81) ; n = 4(9)²(19) = 6,156

So mininum value of n is 396

√(100+√396)+√(100-√396) = (1+√99)+(1-√99) = 2

8×8 square, 1² piece

7×7 square, 2² pieces

6×6 square, 3² pieces

...

1×1 square, 8² pieces

So there are 1²+2²+3²+...+8² = 8(9)(17)/6 = 204 square in chess board.

A geometric series with common ratio between -1 and 1 is a convergent series.

Let 1/2 + 1/4 + 1/8 + 1/16 + ... = S

1/2 + (1/2)( 1/2 + 1/4 + 1/8 + ...) = S

1/2 + (1/2)( S ) = S

S = 1

Problem 17:

Find the value of x that makes x+2, 3x-1, x² arranged in arithmetic sequence.

Problem 18:

Open question to all.

Prove that 2n! / n! • (n+1)! Is always an integer, where n is any positive integer

Problem 17:

Find the value of x that makes x+2, 3x-1, x² arranged in arithmetic sequence.

By definition if the sequence is an arithmetic sequence, x+2+y+y = 3x-1+y = x².

For some y.

2+x+y = 3x-1

=> y = 2x-3

So we have 2+x + 4x -6 = x²

=> x² - 5x +4 = 0

By the quadratic formula, x = 5 ± (25 - 16)½/2

= 5±3/2 = { 4 and 1}

Verifying by substituting the values 4 and 1 for x in the sequence we have-

3,2,1 for x = 1

And

6,11,16 for x = 4

And indeed these things are in arithmetic sequence

Problem 16:

If 4ˣ = 100, find the value of 8ˣ.

(Note : My keyboard can raise things only to the nth power, it is unable to raise to x. So I will substitute n for x)

Note: all logs are to base 4 unless otherwise stated

4ⁿ=100

So n = log(100)

To compute log is tricky.

X+2 = 3x-1+a = x² +2a

X²-4x+6 = x+2

X²-5x+4 =0

X = 5±3/2 = 1 and 4

1 gives 3,2,1 and 4 gives 6,11,16