Math Snap

STEP 1

What is this asking?
We need to break down a complicated fraction into simpler fractions.
Think of it like reversing the addition of fractions!
Watch out!
Don't forget to account for the denominators of the simpler fractions when putting them back together.

STEP 2

1. Factor the Denominator
2. Set up the Decomposition
3. Solve for the Unknowns
4. Verify the Decomposition

STEP 3

Alright, let's factor that denominator!
We're looking for two numbers that multiply to $12 \cdot 10 = 120$ and add up to $-23$.
Those magic numbers are $-8$ and $-15$.

STEP 4

So, we rewrite the quadratic as $12x^2 - 8x - 15x + 10$.
Now, we can factor by grouping.

STEP 5

From the first two terms, we can pull out $4x$, giving us $4x(3x - 2)$.
From the last two terms, we can pull out $-5$, giving us $-5(3x - 2)$.
Notice the common factor $(3x - 2)$!

STEP 6

This means our factored denominator is $(4x - 5)(3x - 2)$.
Boom!

STEP 7

Now, we can set up our partial fraction decomposition:
$$\frac{-53x + 54}{(4x - 5)(3x - 2)} = \frac{A}{3x - 2} + \frac{B}{4x - 5}$$ We're trying to find the mystery numbers $A$ and $B$!

STEP 8

To solve for $A$ and $B$, we'll multiply both sides of the equation by the denominator $(4x - 5)(3x - 2)$.
This gives us:
$$-53x + 54 = A(4x - 5) + B(3x - 2)$$

STEP 9

Let's be clever!
If we set $x = \frac{2}{3}$, the term with $B$ disappears!
$$-53 \cdot \frac{2}{3} + 54 = A(4 \cdot \frac{2}{3} - 5)$$ $$\frac{-106}{3} + \frac{162}{3} = A(\frac{8}{3} - \frac{15}{3})$$$$\frac{56}{3} = A(\frac{-7}{3})$$Multiplying both sides by $3$ and dividing by $-7$ gives us $A = \frac{56}{-7} = -8$.
Yes!

STEP 10

Now, let's set $x = \frac{5}{4}$ to make the term with $A$ vanish!
$$-53 \cdot \frac{5}{4} + 54 = B(3 \cdot \frac{5}{4} - 2)$$ $$\frac{-265}{4} + \frac{216}{4} = B(\frac{15}{4} - \frac{8}{4})$$$$\frac{-49}{4} = B(\frac{7}{4})$$Multiplying both sides by $4$ and dividing by $7$ gives us $B = \frac{-49}{7} = -7$.
Awesome!

STEP 11

Let's verify our solution!
We found $A = -8$ and $B = -7$, so our decomposition is:
$$\frac{-8}{3x - 2} + \frac{-7}{4x - 5}$$

STEP 12

We can rewrite this as a single fraction by finding a common denominator:
$$\frac{-8(4x - 5) - 7(3x - 2)}{(3x - 2)(4x - 5)} = \frac{-32x + 40 - 21x + 14}{12x^2 - 23x + 10} = \frac{-53x + 54}{12x^2 - 23x + 10}$$ It matches the original expression!

The partial fraction decomposition is:
$$\frac{-8}{3x - 2} - \frac{7}{4x - 5}$$