Question

Try this hard Math Problem if you dare!!

Answer 1

a.
`(x - 3)^2 + 16`

b.
`8(x -7)^2`

c.
`(a^2 - 1)(7x - 6)` or
`(a+1)(a-1)(7x-6)`

d.
`(x^2-4)(x^2+3)` or
`(x-2)(x+2)(x^2+3)`

e.
`(a^n+b^n)(a^n-b^n)(a^(2n) +b^(2n))`

Explanation:

`a.\ (x + 1)^2 - 8(x - 1) + 16`

Expand

`(x + 1)(x + 1) - 8(x - 1) + 16`

Open brackets

`x^2 + x + x + 1 - 8x + 8 + 16`

`x^2 + 2x + 1 - 8x + 24`

Collect Like Terms

`x^2 + 2x - 8x+ 1 + 24`

`x^2 - 6x+ 25`

Express 25 as 9 + 16

`x^2 - 6x+ 9 + 16`

Factorize:

`x^2 - 3x - 3x + 9 + 16`

`x(x -3)-3(x - 3) + 16`

`(x - 3)(x - 3) + 16`

`(x - 3)^2 + 16`

`b.\ 8(x - 3)^2 - 64(x-3) + 128`

Expand

`8(x - 3)(x - 3) - 64(x-3) + 128`

`8(x^2 - 6x+ 9) - 64(x-3) + 128`

Open Brackets

`8x^2 - 48x+ 72 - 64x+192 + 128`

Collect Like Terms

`8x^2 - 48x - 64x+192 + 128+ 72`

`8x^2 -112x+392`

Factorize

`8(x^2 -14x+49)`

Expand the expression in bracket

`8(x^2 -7x-7x+49)`

Factorize:

`8(x(x -7)-7(x-7))`

`8((x -7)(x-7))`

`8(x -7)^2`

`c.\ 7a^2x - 6a^2 - 7x + 6`

Factorize

`a^2(7x - 6) -1( 7x - 6)`

`(a^2 - 1)(7x - 6)`

The answer can be in this form of further expanded as follows:

`(a^2 - 1^2)(7x - 6)`

Apply difference of two squares

`(a+1)(a-1)(7x-6)`

`d.\ x^4 - x^2 - 12`

Express
`x^4` as
`x^2`

`(x^2)^2 - x^2 - 12`

Expand

`(x^2)^2 +3x^2- 4x^2 - 12`

`x^2(x^2+3) -4(x^2+3)`

`(x^2-4)(x^2+3)`

The answer can be in this form of further expanded as follows:

`(x^2-2^2)(x^2+3)`

Apply difference of two squares

`(x-2)(x+2)(x^2+3)`

`e.\ a^(4n) -b^(4n)`

Represent as squares

`(a^(2n))^2 -(b^(2n))^2`

Apply difference of two squares

`(a^(2n) -b^(2n))(a^(2n) +b^(2n))`

Represent as squares

`((a^(n))^2 -(b^(n))^2)(a^(2n) +b^(2n))`

Apply difference of two squares

`(a^n+b^n)(a^n-b^n)(a^(2n) +b^(2n))`