Question

Find all the roots of the following equationsx^3-2x^2-5x+10=0 using the p and q method

Answer 1

EXPLANATION

Given the equation:

x^3 -2x^2 -5x + 10 = 0

Grouping terms:

`=(x^3-2x^2)+(-5x+_{}10)`

Factor out -5 from (-5x+10) and x^2 from (x^3-2x^2):

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

Factor out common term x-2:

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

Factor x^2-5:

`x^2-5=x^2-(\sqrt[]{5})^2`

Apply difference of two squares formula:

`x^2-y^2=(x+y)(x-y)`
`x^2-(\sqrt[]{5})^2=(x+\sqrt[]{5})(x-\sqrt[]{5})`

Grouping factor:

`=(x-2)(x+\sqrt[]{5})(x-\sqrt[]{5})`

Equaling to zero in order to get the roots:

`(x-2)(x+\sqrt[]{5})(x-\sqrt[]{5})=0`

Using the zero factor principle:

`\text{If ab=0, then a=0 or b=0}`
`x-2=0\longrightarrow\text{ x=2 \lbrack{}First root\rbrack}`
`x+\sqrt[]{5}=0\longrightarrow\text{ x=-}\sqrt[]{5}\text{ \lbrack{}Second root\rbrack}`
`x-\sqrt[]{5}=0\longrightarrow\text{ x=}\sqrt[]{5}\lbrack Third\text{ Root\rbrack}`

The roots are:

x=2, x=-sqrt(5) and x=sqrt(5)