Question

Solve the following? `(x−1)(x−3)(x−5)(x+1)=−12`

Answer 1

`x=−0.645751,0.267949,3.732051,4.645751`

Explanation:

`(x-1)(x-3)(x-5)(x+1)=-12`

first, we can expand each of the brackets using FOIL

`(x-1)(x-3) = x^2 - 3x - x + 3`

`(x-5)(x+1) = x^2 +x -5x - 5`

Now we can simplify these and combine them back together into the original expression.

`(x^2 - 4x + 3) xx (x^2 -4x - 5) = -12`

Now we can factorise this.

`(color(lime)(x^2 - 4x + 3)) (color(skyblue)(x^2 -4x - 5)) = (color(lime)(x^2)) (color(skyblue)(x^2)) + (color(lime)(x^2))(color(skyblue)(-4x)) + (color(lime)(x^2))(color(skyblue)(-5)) + (color(lime)(-4x))(color(skyblue)(x^2)) + (color(lime)(-4x))(color(skyblue)(-4x)) + (color(lime)(-4x))(color(skyblue)(-5)) + (color(lime)(3))(color(skyblue)(x^2)) + (color(lime)(3))(color(skyblue)(-4x)) + (color(lime)(3))(color(skyblue)(-5))`

`= (x^4) + (-4x^3) + (-5x^2) + (-4x^3) + (16x^2) + (20x) + (3x^2) + (-12x) + (-15)`

Now we can collect like terms and simplify.

`= x^4 -4x^3 - 4x^3 -5x^2 + 3x^2 + 16x^2 + 20x -12x -15`

`x^4 - 8x^3 + 14x^2 + 8x -15 = -12`

Now we can find `x`

`x^4 - 8x^3 + 14x^2 + 8x color(red)(cancel(color(black)(-15) +15)) = -12 color(red)(+15)`

`x^4 - 8x^3 + 14x^2 + 8x = 3`

The only way that I could find to solve this is through using the Quartic formula.

`color(blue)(x=−0.645751,0.267949,3.732051,4.645751)`

Answer 2

`+-sqrt3+2` and `+-sqrt7+2`

Explanation:

We have:

`(x-1)(x-3)(x-5)(x+1)=-12`

Multiply the first two and the last two parentheses together.

`=>(x^2-4x+3)(x^2-4x-5)=-12`

Hmm... We see that the first quadratic expression is 8 larger than the second.

We let `s=(x^2-4x+3)`

We now have:

`s(s-8)=-12`

`=>s^2-8s=-12` complete the square

`=>(s^2-8s+16)-16=-12`

`=>(s-4)^2=4`

`=>-2=s-4=2`

`=>2=s=6`

We can use this to solve for `x`.

When `s=2`...

`x^2-4x+3=2`

`=>x^2-4x+1=0`

`=>(x^2-4x+4)-4+1=0`

`=>(x-2)^2-3=0`

`=>(x-2)^2=3`

`=>x-2=+-sqrt(3)`

`=>x=+-sqrt(3)+2`

When `s=6`...

`x^2-4x+3=6`

`=>x^2-4x-3=0`

`=>(x^2-4x+4)-4-3=0`

`=>(x-2)^2-7=0`

`=>(x-2)^2=7`

`=>x-2=+-sqrt(7)`

`=>x=+-sqrt(7)+2`

The answers are `+-sqrt3+2` and `+-sqrt7+2`