Factorizing Quadratic Equation: 4x^2 - 12x - 14 | Step-by-Step Solution

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In summary, the conversation discusses how to completely factorize the expression 4x^2 - 12x - 14 and whether it can be simplified further. The quadratic formula is used to find the roots, which are irrational numbers. The final factorization is (3 ± √23)/2.
  • #1
Learnphysics
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Homework Statement



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

Homework Equations


The Attempt at a Solution



Quadratic equation:
(-(-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?
 
Last edited:
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  • #2
[tex](4x\pm a)(x\pm b)[/tex]

or

[tex](2x\pm a)(2x\pm b)[/tex] <--- 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.
 
Last edited:
  • #3
The quadratic equation finds the roots of [itex]ax^2+bx+c=0, a\neq 0[/itex], and the factor theorem says that if [itex]P(a) = 0, (x-a) [/itex] is a factor of P(x). So finding the zeros will enable to factor that quadratic.
 
  • #4
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))
 
  • #5
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.
 
  • #6
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)(...)
 
  • #7
... strange that the spaces did not get filled in on one of the lines shown above.
 
  • #8
apparently, none of my four suggested combinations will work. Someone else check? Typographical error in given expression?
 
  • #9
Hi,
The question is Completely factorize [tex]4x^2 - 12x - 14[/tex]
I used the quadratic formula and got:
[tex](x+\frac{\sqrt23+3}{2})(x-\frac{\sqrt23+3}{2})[/tex]
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
 
  • #10
happyg1 said:
Hi,
The question is Completely factorize [tex]4x^2 - 12x - 14[/tex]
I used the quadratic formula and got:
[tex](x+\frac{\sqrt23+3}{2})(x-\frac{\sqrt23+3}{2})[/tex]
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,
 
  • #11
… the straight answer …

Learnphysics said:
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! :smile:

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. :smile:

(btw, you must put the ± into your (12+ sqrt(368))/8)
 

What is factorizing a quadratic equation?

Factorizing a quadratic equation is the process of breaking down a quadratic equation into simpler expressions, known as factors. These factors are multiplied together to get the original quadratic equation.

Why is it important to factorize a quadratic equation?

Factorizing a quadratic equation helps in solving the equation easily and finding the roots or solutions. It also helps in graphing the equation and understanding its properties.

What is the general method for factorizing a quadratic equation?

The general method for factorizing a quadratic equation is to find two numbers that multiply to give the constant term in the equation and add to give the coefficient of the middle term. These two numbers will be the factors of the quadratic equation.

How do you factorize the quadratic equation 4x^2 - 12x - 14?

To factorize the quadratic equation 4x^2 - 12x - 14, we first find two numbers that multiply to give -56 (product of the constant term -14) and add to give -12 (coefficient of the middle term). These numbers are -14 and 4. We then rewrite the middle term as -14x + 4x and factorize the equation as 4x(x-3)-14(x-3), which gives us the factors (4x-14)(x-3).

What are the steps involved in factorizing a quadratic equation?

The steps involved in factorizing a quadratic equation are:

  1. Write the equation in the form ax^2 + bx + c = 0
  2. Find two numbers that multiply to give ac (product of the coefficients of x^2 and c) and add to give b (coefficient of x)
  3. Rewrite the middle term as the sum of these two numbers
  4. Factorize the equation using the distributive property
  5. Write the factors in the form (px+q)(rx+s)

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