# Homework Help: How To Find The Factors Of THis Quadratic Equation?

1. Jul 12, 2012

### optics.tech

Hi everyone,

Can someone please tell me how to find the factors of the below quadratic equation?

$$3x^2 - 9x + 4 = 0$$

I had already tried to find them by using the below method and wasn't able to continue further because of I do not understand.

Huygen

$$(3x^2 - 7x + 4) -2x = 0$$
$$(3x-4)(x-1)-2x=0$$

2. Jul 12, 2012

### DonAntonio

Since $\,\Delta:=b^2-4ac=9^2-4\cdot 3\cdot 4=33\,$ , this quadratic's roots are ugly:
$$x_{1,2}=\frac{-b\pm\sqrt\Delta}{2a}=\frac{9\pm\sqrt{33}}{6}$$ , so we can now factor
$$3x^2-9x+4=3\left[x-\left(\frac{9-\sqrt{33}}{6}\right)\right]\left[x-\left(\frac{9+\sqrt{33}}{6}\right)\right]$$
Ugly, indeed.

DonAntonio

3. Jul 13, 2012

### optics.tech

No, not this method.

There is available another method than this one.

4. Jul 13, 2012

### D H

Staff Emeritus
Completing the squares.

5. Jul 13, 2012

### eumyang

Are you looking for the factors, or are you looking for the roots? If you are looking for factors, and they have to have integer coefficients, then you won't find any, as DonAntonio has shown. If you don't want to use the quadratic formula (again, as DonAntonio has shown), you'll need to complete the square, which D H suggested.

6. Jul 13, 2012

### Mentallic

In comparison, if you use the quadratic formula on the separate quadratic you factorized:

$$(3x^2 - 7x + 4) -2x = 0$$
$$(3x-4)(x-1)-2x=0$$

So we're looking at $3x^2-7x+4$ only, then the discriminant

$$\Delta=b^2-4ac$$$$=(-7)^2-4(3)(4)$$$$=49-48=1$$

and thus since the square root of that is a rational number, then you can factorize the quadratic using only integer coefficients as you've shown.

7. Jul 13, 2012

### optics.tech

Yes, you are correct.