Algebra question?

Question

I don't know if I understood what "roots" are correctly.

The problem was to compute the sum of all the roots of
(2x + 3)(x - 4) + (2x+3)(x - 6) = 0

I figured x = -1.5
= 4
= 6

and the answer I got was 8.5

Can someone tell me if I got this problem right? and if it's wrong, why? Thanks!

Answer 1

(2x + 3) (x - 4 + x - 6) = 0
(2x + 3)(2x - 10) = 0
x = - 3/2 , x = 5

Answer 2

Well, for one thing, neither 4 nor 6 are roots of the equation. Look at what happens when you plug in 4:

(2(4) + 3)(4-4) + (2(4) + 3)(4-6) = 0 + 11(-2) = -22 ≠ 0.

Your error stems in a misapplication of the zero-product property -- the correct version states that if x*y = 0, then either x=0 or y=0. It does NOT state that if x+y = 0 then either x=0 or y=0. Nor does it state that if x*y + x*z = 0, then either x=0, y=0, or z=0 (which is what you seem to have assumed). Before you can use it, you must transform this into a PRODUCT of smaller factors, not a SUM of such factors.

Now, we start with:

(2x + 3)(x - 4) + (2x+3)(x - 6) = 0

Note that one of the factors in each term is the same, so we can use the distributive property to combine them:

(2x + 3)((x - 4) + (x - 6)) = 0

Simplifying:

(2x + 3)(2x - 10) = 0

NOW, we can use the zero-product property. We have:

2x+3 = 0 ∨ 2x - 10 = 0
2x = -3 ∨ 2x = 10
x = -3/2 ∨ x=5

So the roots are -3/2 and 5, and their sum is 7/2.

Answer 3

No, not quite - roots are numbers which if you plug them into the equation for x make the entire thing equal zero. If you put in x equals 4 or 6, you don't get zero. The reason is that you haven't reduced the equation to products yet by collecting like terms. Taking out a factor of (2x+3) gives:

(2x+3)(x-4+x-6)=0
(2x+3)(2x-10)=0
(2x+3)(x-5)=0

So the roots are -1.5 and 5, and the sum is -3.5

You have to have a product of terms only to find the roots, otherwise as shown with the 4/6 example above, you can have non-zero solutions, which means those numbers aren't roots.

Good luck!

Answer 4

4 and 6 are not roots. Substitute 4 for x and you'll see why.

(2x + 3)(x - 4) + (2x+3)(x - 6) = 0
(2x + 3)[(x - 4) + (x - 6)] = 0

2x² - 8x + 3x - 12 + 2x² - 12x + 3x - 18 = 0
(2x + 3)[2x - 10] = 0

4x² - 14x - 30 = 0
x² - (7/2)x - (15/2) = 0
x² - (7/2)x = 15/2
Complete the square:
x² - (7/2)x + (7/4)² = 15/2 + (7/4)²
(x - 7/4)² = (120+49)/16
(x - 7/4) = ±√(169/16)
= (7/4)±[√169/√16]
= (7/4)±(13/4)
x₊ = 20/4 = 5
x₋ = -6/4 = -1.5

Answer 5

If you plug 4 or 6 into the original equation, you don't get a zero answer. The equation isn't right.

I think you need to simplify first:

(2x + 3)(x-4) + (2x + 3)(x-6) = (2x+3)(2x-10) = 0

The roots would therefore be 5 and -1.5, for an answer of 3.5.

Answer 6

First off, you can't have three answers for x, because if you multiply everything out and simplify, the highest power of x you have is 2, meaning there can be 2 or less roots.

ok, so do this, you start by factoring. (2x+3) is a common factor, so factor that out to get:
(2x+3)((x-4)+(x-6))=0 which is the same as
(2x+3)(2x-10)=0

So x= -3/2 = -1.5
or x=5

so the sum is 5-1.5=3.5

I think you didn't simplify (x-4)+(x-6) and thought those were also factors, but they're not because you are adding them, not multiplying them.

Answer 7

Hi there,

the equation u wrote is correct and can be solve like that:

i) By multiplying brackets,The equation will be:
2x²-5x-12+2x²-9x-18=0

ii)The equation now is:
4x²-14x-30=0

iii)Dividing the whole equation on 2,We will have:
2x²-7x-15=0

iv)Analysing:
(2x+3)(x-5)=0

so, 2x+3=0 and x-5=0

so x=-1.5 and x=5

for further info pls tell me
If my solution is not right pls tell me

Good luck,

Aquabigo
Shoubra Faculty of engineering
Fisrt year
Benha university
Egypt

Answer 8

I think thats right thats what i would have got.