Simplify $9((x^2-15x+50)/84)-12((x^2-8x-20)/-35)+33((x^2-3x-10/60)$

Question

Hi I am trying to simplify the following I found online

$9\left(\dfrac{x^2-15x+50}{84}\right) + -12\left(\dfrac{x^2-8x-20}{-35}\right) + 33\left(\dfrac{x^2-3x-10}{60}\right)$

to

$= x^2 -6x -7$

Working through this I thought I could multiply the top by the divide for each part of the polynomial. For example

$x^2 \cdot 84 = 84x^2$

$-15x \cdot 84 = 1260x$

$+50 \cdot 84 = 4200$

Then to remove the outside multiply times again.

$84x^2 \cdot 9 = 756x^2$

$-1260x \cdot 9 = 11340x$

$4200 \cdot 9 = 37800$

Do this for all of the parts and then simplify down to the expected

$= x^2-6x-7$

It did not seem to work though unless I was missing something or made a mistake.

Is this the expected method of doing this?

I came up with $3156x^2 - 20640x - 27420$

But I cannot see how that would simplify to the expected.

Answer 1

To simplify $9\left(\dfrac{x^2-15x+50}{84}\right) + -12\left(\dfrac{x^2-8x-20}{-35}\right) + 33\left(\dfrac{x^2-3x-10}{60}\right),$

first simplify to $$3\left(\dfrac{x^2-15x+50}{28}\right) + 12\left(\dfrac{x^2-8x-20}{35}\right) + 11\left(\dfrac{x^2-3x-10}{20}\right).$$

Now give the fractions their lowest common denominator:

$$15\left(\dfrac{x^2-15x+50}{140}\right) + 48\left(\dfrac{x^2-8x-20}{140}\right) + 77\left(\dfrac{x^2-3x-10}{140}\right).$$

Now the fractions can be added: $$\dfrac{15\left({x^2-15x+50}\right) + 48\left({x^2-8x-20}\right) + 77\left({x^2-3x-10}\right)}{140}.$$

Can you conclude?