Question

What is the standard form of `y= (4x-15)(2x-2)-(3x-1)^2`?

Answer 1

`y = -x^2 - 32x + 29`

Here's how I did it:

Explanation:

Standard form means that we have to put the equation in this form: `y = ax^2 + bx + c`.

`y = (4x-15)(2x-2)-(3x-1)^2`

The first thing we have to do is distribute and expand:
`4x * 2x = 8x^2`

`4x * -2 = -8x`

`-15 * 2x = -30x`

`-15 * -2 = 30`

When we combine this all together we get:
`8x^2 - 8x - 30x + 30`

We can still combine like terms by doing `-8x - 30x`:
`8x^2 - 38x + 30`

`-------------------`

Now let's look at `(3x-1)^2` and expand:
`(3x-1)(3x-1)`

`3x * 3x = 9x^2`

`3x * -1 = -3x`

`-1 * 3x = -3x`

`-1 * -1 = 1`

When we combine this all together we get:
`9x^2 - 3x - 3x + 1`

Then we combine like terms by doing `-3x-3x`:
`9x^2 - 6x + 1`

`------------------`

So the equation is now:
`y = 8x^2 - 38x + 30 - (9x^2 - 6x + 1)`

Let's distribute the negative sign:
`y = 8x^2 - 38x + 30 - 9x^2 + 6x - 1`

Finally, let's combine like terms again:
`y = color(red)(8x^2) quadcolor(magenta)(-quad38x) + color(blue)30 quadcolor(red)(-quad9x^2) + color(magenta)(6x) quadcolor(blue)(-quad1)`

So the final answer in standard form is:
`y = -x^2 - 32x + 29`
as it matches `y = ax^2 + bx + c`.

Hope this helps!