Help on composite functions with a quadratic

Neither the MathsWatch nor MathsGenie videos deal with composite functions with two answers (quadratic equations) so I do not get what's going on in this MathsGenie paper to get the last two lines:

Screenshot 2024-06-06 at 21-33-08 7-functionsans.pdf.png

Screenshot 2024-06-06 at 21-33-08 7-functionsans.pdf.png

Could anyone explain why 0 = 6x(x+1) becomes x=0,x=-1?

(PS. very stupid that the only way to add an image to a post in this "improved" design is to edit it in after cause there are literally no rich text functions, idiotic UX)

(edited 1 year ago)

Reply 1

miffy122233

when you solve for x the x value equates to -1 and anything with a single bracket so 6x(x+1) will have a 0 value that's just the rule.

Reply 2

The_Architect

Original post by miffy122233

when you solve for x the x value equates to -1 and anything with a single bracket so 6x(x+1) will have a 0 value that's just the rule.

Is this only when doing composite functions or for all quadratic equations? I haven't seen this mentioned in any video I've seen

(edited 1 year ago)

Reply 3

davros

Original post by The_Architect

Is this only when doing composite functions or for all quadratic equations? I haven't seen this mentioned in any video I've seen

Either I'm being dumb, or you're over-thinking this!

You have an equation like x(x+1) = 0 (the 6 is irrelevant). If the product of 2 numbers is 0, then one of the numbers has to be 0. So either x = 0 or x+1 = 0 i.e. x = -1.

Or are you asking about something else?

Reply 4

lvnrn

if

0 = 6x(x + 1)

, then either 6x AND/OR (x + 1) must equal 0 in order for them to be multiplied together to make zero.

say 6x = 0 to demonstrate.

if 6x = 0, then therefore 6x(x + 1) can be replaced with (0)(x + 1), which means we can literally just ignore the (x + 1) since it is multiplied by zero. and the same works the other way around:

if (x + 1) = 0, therefore 6x(x + 1) = 6x(0) = 0

that means we can conlude that either 6x = 0 or (x + 1) = 0 or both.

now if 6x = 0 then obviously x = 0.

if x + 1 = 0 then x = -1.

That means that EITHER x = 0 OR x = -1 (or sometimes both, depends on context of question if applicable).

Also, with quadratic equations, in their factorised form (e.g. x² - x - 20 can be re-written as (x - 5)(x + 4) ), show their roots.

therefore the roots of this quadratic are x = 5 and x = -4 --> at these points in the curve, y = 0.

Reply 5

The_Architect

Thanks to you both, I remembered after that quadratic equations need to have both sides of the equation solved and since 0 is already solved you just solve for the remaining side.

(edited 1 year ago)

Reply 6

davros

Original post by The_Architect

Thanks to you both, I remembered after that quadratic equations need to have both sides of the equation solved and since 0 is already solved you just solve for the remaining side.

What do you mean by "both sides of the equation solved"? That doesn't make any sense - you solve an equation, you don't "solve" each side separately!