Question

X⁴+4x³-3x²-14x=8
find the root for this quadratic equation SHOW YOUR WORK PLEASE

Answer 1

Hi !

Let's resolve this equation :

`x^(4)+4x^(3)-3x^(2) -14x=8\\x^(4)+4x^(3)-3x^(2) -14x-8=0`

We can see that
`x_(1)=-1` is an obvious root of the polynomial

`x^(4)+4x^(3)-3x^(2) -14x-8`.

So we can divide this one by
`x-(-1)=x+1`.

See attached.

So,
`x^(4)+4x^(3)-3x^(2) -14x-8=(x+1)(x^(3)+3x^(2) -6x-8)`

We can see that
`x_(2)=2` is an obvious root of the polynomial
`x^(3)+3x^(2) -6x-8`.

So we can divide
`x^(3)+3x^(2) -6x-8` by
`x-2`.

See attached.

So,
`x^(3)+3x^(2) -6x-8=(x-2)(x^(2) +5x+4)`

So,
`\boxed{x^(4)+4x^(3)-3x^(2) -14x-8=(x+1)(x-2)(x^(2) +5x+4)}`

The discriminant of
`x^(2) +5x+4` is :

`\Delta=5^(2)-4* 1* 4=25-16=9` and
`√(\Delta)=3 \ or \ √(\Delta)=-3`

We have two roots :

`x_(3)=(-5-3)/(2)=-4 \\\\x_(4)=(-5+3)/(2)=-1=x_(1)`

So all the roots of the polynomial
`x^(4)+4x^(3)-3x^(2) -14x-8` are -4, -1 and 2.

;-)