factorize fully:
4x^2 - 12x - 14

## The Attempt at a Solution

(-(-12)+ sqrt(12^2 - (4 * 4 * -14))/ 2 * 4
(12 + sqrt(144 + 224))/8

in the end it became a surd
(12+ sqrt(368))/8

Could someone tell me how to factorize this?
or is that as far as it will go?

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Related Precalculus Mathematics Homework Help News on Phys.org
$$(4x\pm a)(x\pm b)$$

or

$$(2x\pm a)(2x\pm b)$$ <--- Try this one.

The sign of the middle term tells who the sign of the bigger value, and the last term tells you whether or not you will have 2 positive/negative values or a positive & negative value.

If all fails, you may need the quadratic equation.

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Gib Z
Homework Helper
The quadratic equation finds the roots of $ax^2+bx+c=0, a\neq 0$, and the factor theorem says that if $P(a) = 0, (x-a)$ is a factor of P(x). So finding the zeros will enable to factor that quadratic.

Is there a typo? Because 4x^2-10x-14 factors very nicely...
If your teachers are trying to throw you a curve ball, then you must use the quadratic equation, with zeroes (1.5+sqrt(92/16)) and (1.5- sqrt(92/16))

What I did was using the solution of this quadratic when equated to zero using the quadratic formula & this is what I got;

(x - 3/2 - 1/2*sqrt(14))*(x - 3/2 + 1/2*sqrt(14))

I haven't checked it so maybe an error in my calculations,but I don't really think so...

Thats the technique I use for all my quadratic factorizations & its simply the easiest.

symbolipoint
Homework Helper
Gold Member
First, do you see a factor of 2 which can be factored:

2(2x^2 - 6x -7)

Now, you can plan for two binomial factors to look for the other terms within each binomial:

(2x )(x )

Look for different ways of getting a product of -7, so use like:

(2x -7 )(x +1 )
OR
(2x +7)(x -1 )
OR
(2x -1)(x +7)
OR
(you fill in the rest)(.....)

symbolipoint
Homework Helper
Gold Member
... strange that the spaces did not get filled in on one of the lines shown above.

symbolipoint
Homework Helper
Gold Member
apparantly, none of my four suggested combinations will work. Someone else check? Typographical error in given expression?

Hi,
The question is Completely factorize $$4x^2 - 12x - 14$$
I used the quadratic formula and got:
$$(x+\frac{\sqrt23+3}{2})(x-\frac{\sqrt23+3}{2})$$
The expression does not factor nicely. It's "prime" There are no combinations that symbolipoint suggested that will work and give nice pretty integers. No factors of 14 subtract to give you 6.
CC

Hi,
The question is Completely factorize $$4x^2 - 12x - 14$$
I used the quadratic formula and got:
$$(x+\frac{\sqrt23+3}{2})(x-\frac{\sqrt23+3}{2})$$
The expression does not factor nicely. It's "prime" There are no combinations that symbolipoint suggested that will work and give nice pretty integers. No factors of 14 subtract to give you 6.
CC
Don't forget to multiply with 4 again. 4x^2 - 12x - 14 has the same roots as
x^2 - 3x - 7/2 but not the same factorization,

tiny-tim
Homework Helper

in the end it became a surd
(12+ sqrt(368))/8

Could someone tell me how to factorize this?
or is that as far as it will go?
Hi Learnphysics! The straight answer to your question is that you can't get rid of the square root, though you can simplify it.

As happyg1 almost says, it's (3 ± √23)/2. (btw, you must put the ± into your (12+ sqrt(368))/8)