Question

Solve the equation 2x3 + 7x2 – 10x – 24 = 0 in the real number system.

Answer 1

x = 2

x = -4

x = -3/2

Explanation:

We need to find the roots of the equation

2x^3 + 7x^2 – 10x – 24 = 0

We can factorize the equation into

(x-2)(x+4)(2x + 3) = 0

Roots

x = 2

x = -4

x = -3/2

The answer to your question is also attached in the images below

Answer 2

x=2 ; x=-4 or x=-3/2

Explanation:

We are given a cubic polynomial

`2x^3+7x^2-10x-24=0`

Here as we are having a cubic equation to factorize, we have to find some value of x such that when we put that value in place of x , it gives us a 0. We have to try it randomly like first we try it for x=1 then x=2 and so on.

On trying for x=2 , we see that the equation becomes 0=0 and hence x=2 is one of the solution. And also as x=2 is one of the solution, (x-2) must be one of the factor. Now we use this property to determine the other factor like this.

`2x^3+7x^2-10x-24=0`

Adding and subtracing
`4x^2` to
`2x^3` we get

`2x^3-4x^2+4x^2+7x^2-10x-24=0`

Now we take out
`2x^2` as GCF

`2x^2(x-2)+11x^2-10x-24=0`

Now subtracting and adding
`22x` to
`11x^2`

`2x^2(x-2)+11x^2-22x+22x-10x-24=0`

taking
`11x^2` as GCF out

`2x^2(x-2)+11x(x-2)+22x-10x-24=0`

`2x^2(x-2)+11x(x-2)+12x-24=0`

`2x^2(x-2)+11x(x-2)+12(x-2)=0`

Now taking
`(x-2)` out as GCF

`(x-2)(2x^2+11x+12)`

Now we need to split the middle term of
`(2x^2+11x+12)` to factorize it in such a way that their product is 2*12 and sum is 11. We have 8 and 3 as these factors.

Hence it can be factored like this

`(2x^2+11x+12)\\2x^2+8x+3x+12\\2x(x+4)+3(x+4)\\(2x+3)(x+4)`

Hence our answer will be

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

And thus the solutions will be

x-2=0 : x=2

x+4=0 ; x=-4

2x+3=0 ; x=-
`(3)/(2)`