# How do you factor 6x^2-5x-25?

May 6, 2018

The answer is: $\left(2 x - 5\right) \left(3 x + 5\right)$

#### Explanation:

So factoring may seem hard but let take a look at what we could do.

So you first think of factors of the coefficient in front of $6 {x}^{2}$. Now there are a couple of terms that do get us to six by multiplying but it should also add to the middle term.

Now, if I choose $6$ and $1$, that doesn't work because it wouldn't match the middle term. If I choose $2$ and $3$, it would work . because it works for $a$ and $b$ (Standard form is: $a x + b y = c$)

So let put it into the equation. But before we do that, we need a number that going to work for $- 25$ which is positive and negative $5.$ You will see why we need it.

$\left(2 x - 5\right) \left(3 x + 5\right)$
$6 {x}^{2} + 10 x - 5 x - 25$

Works :D

May 6, 2018

x = $- \frac{5}{3}$ or $\frac{5}{2}$

#### Explanation:

$6 {x}^{2} - 5 x - 25$ = 0

Using the umbrella-XBOX method:

Multiply $- 25$ by $6 {x}^{2}$.

You should now have:

$- 150 {x}^{2}$ and $- 5 x$ = 0

You must figure out what multiplies to get $- 150 {x}^{2}$ and adds to $- 5 x$. If you use a factoring tree for 150, it'll help you find your answer.

1 - 150
2 - 75
3 - 50
5 - 30
6 - 25
10 - 15
15 - 10

25 - 6
30 - 5
50 - 3
75 - 2
150 - 1

$- 15 x$ and $10 x$ multiply to give you $- 150 {x}^{2}$, but also add to give you $- 5 x$. Rewrite your equation with your 2nd term broken apart into $- 15 x$ and $10 x$:

$6 {x}^{2} - 15 x + 10 x - 25$ = 0

Make a square box with four sections. Put the first term in the first box, the last term last, with the two middle terms in the two middle boxes. Then, do it like a Punnett square: Here's how I've arranged mine, left to right: BB is $6 {x}^{2}$, Bb is 10x, the other Bb is -15x, and bb is -25. Now, 2x and 3x multiply to make $6 {x}^{2}$, so the paternal B will be 2x (to make it easier with the 10x on right) and the maternal B will be 3x. Now, because 2x multiplied by something makes 10x, the maternal B is 3x and the maternal b is 5.

The next Bb is -15x. The maternal B is 3x, so 3x has to multiply by something to become -15x. -15x/3x = -5 , so the paternal b will be -5.

Therefore, maternal B is 3x, maternal b is 5, paternal B is 2x, and paternal b is -5, which is written as:

(3x + 5) = 0
(2x - 5) = 0

Solve like a one-step equation and you get:

x = $- \frac{5}{3}$ or $\frac{5}{2}$

BETTER EXPLANATION:

SOURCE for PUNNETT SQUARE DIAGRAM:
https://en.wikipedia.org/wiki/Reginald_Punnett

May 7, 2018

(3x + 5)(2x - 5)

#### Explanation:

Use the new AC Method (Socratic Search)
$y = 6 {x}^{2} - 5 x - 25$.
Converted trinomial:
$y ' = {x}^{2} - 5 x - 150$
Proceeding. Find the factor numbers of y', then, divide them by
a = 6. Find 2 numbers, that have opposite signs (ac < 0), knowing their sum (b = -5) and their product (ac = - 150).
They are: 10 and - 15.
The factor numbers of y are: $\frac{10}{a} = \frac{10}{6} = \frac{5}{3}$, and $- \frac{15}{6} = - \frac{5}{2}$.
Factored form:
$y = 6 \left(x + \frac{5}{3}\right) \left(x - \frac{5}{2}\right) = \left(3 x + 5\right) \left(2 x - 5\right)$
Note. This method avoids the lengthy factoring by grouping, and the solving of the 2 binomials.