Question

How many solutions can f (x) = g (x) have if f and g are both quadratic functions?
1. The equation can have 0, 1, or 2 solutions because two parabolas may fail to intersect, may intersect exactly once, or may intersect twice.
2. They can have 0, 1, 2, or infinitely many solutions, depending on whether the two functions intersect 0, 1, or 2 times, or infinitely many if they happen to be the same function.
3. The equation will always have either 1 or 2 solutions depending on whether f and g intersect once or twice, but they always intersect at least once.
4. The equation will have 0, 2, or infinitely many solutions because the parabolas either never intersect, intersect twice, or they are the same parabola.

Answer 1

What do you think the answer is, and why?

Answer 2

\(\color{#0cbb34}{\text{Originally Posted by}}\) @Extrinix
What do you think the answer is, and why?
\(\color{#0cbb34}{\text{End of Quote}}\)
idk thats why i posted it on here

Answer 3

Hello @dakchicken

do you know what a quadratic function is and what it looks like when its graphed?

Answer 4

\(\color{#0cbb34}{\text{Originally Posted by}}\) @AZ
Hello @dakchicken

do you know what a quadratic function is and what it looks like when its graphed?
\(\color{#0cbb34}{\text{End of Quote}}\)
kinda not that much

Answer 5

so a quadratic function is something that has x^2

it could be

x^2 + 5x + 6
or
x^2 -16

Answer 6

but basically when you graph it, it looks like a parabola

Does a graph like this look familiar to you?

Answer 7

\(\color{#0cbb34}{\text{Originally Posted by}}\) @AZ
but basically when you graph it, it looks like a parabola

Does a graph like this look familiar to you?

Created with RaphaëlReply Using Drawing
\(\color{#0cbb34}{\text{End of Quote}}\)
yes

Answer 8

okay, so now to answer your question, we need to imagine we have 2 different parabolas like this

and we're looking for how many times they cross over

Answer 9

so one example is

how many times do the two curves intersect each other?
or cross each other?

Answer 10

once

Answer 11

Good! Now that was one example but what if the second quadratic function was something like

now how many times do they cross each other?

Answer 12

twice

Answer 13

good! So now we know that depending on the graphs, we can have either 1 solution or 2 solutions

but what if we also have, now how many times does it intersect each other?

Answer 14

whts the answer to my question tho

Answer 15

we're almost there!! how many times does it intersect? Wouldn't you say infinitely many times?

Answer 16

3

Answer 17

well, it's way more than three

I tried to draw the line on top of it so basically it would cross the line at every single point which would be too many to count so we would say it's infinitely many times

Answer 18

5

Answer 19

now we have one last scenario and that's if we have something like

how many times do these two graphs cross each other?

Answer 20

0

Answer 21

exactly so what do you think the final answer is?

we just said it could have 0, 1, 2 or infinitely many solutions

Answer 22

\(\color{#0cbb34}{\text{Originally Posted by}}\) @AZ
okay, so now to answer your question, we need to imagine we have 2 different parabolas like this

and we're looking for how many times they cross over
\(\color{#0cbb34}{\text{End of Quote}}\)
@AZ given f(x)=g(x) so then how will they parabolas will intersect ? i think has the same parabolas both two quadratics bc. are equal

Answer 23

f (x) = g (x) have if f and g are both quadratic functions

given this - yes ?

Answer 24

that dont answer it

Answer 25

from this result if f(x) = x^2 +3x -10 so then g(x) = x^2 +3x -10 too

@AZ please

Answer 26

so what result from this ?

Answer 27

we know that a quadratic has two roots - so then ?

Answer 28

yes depend the discriminant is greater then zero or if equal zero then x_1 = x_2 so has two equal roots or if the discriminant is smaller then zero havent real roots

Answer 29

\(\color{#0cbb34}{\text{Originally Posted by}}\) @dakchicken
that dont answer it
\(\color{#0cbb34}{\text{End of Quote}}\)



read what I said and then take a look at your answer choices



\(\color{#0cbb34}{\text{Originally Posted by}}\) @AZ
exactly so what do you think the final answer is?

we just said it could have 0, 1, 2 or infinitely many solutions
\(\color{#0cbb34}{\text{End of Quote}}\)

Answer 30

\(\color{#0cbb34}{\text{Originally Posted by}}\) @AZ
\(\color{#0cbb34}{\text{Originally Posted by}}\) @dakchicken
that dont answer it
\(\color{#0cbb34}{\text{End of Quote}}\)



read what I said and then take a look at your answer choices



\(\color{#0cbb34}{\text{Originally Posted by}}\) @AZ
exactly so what do you think the final answer is?

we just said it could have 0, 1, 2 or infinitely many solutions
\(\color{#0cbb34}{\text{End of Quote}}\)
\(\color{#0cbb34}{\text{End of Quote}}\)
2

Answer 31

That's correct

Answer 32

or like an other idea you can consider f(x)=g(x) a system of quadratics

f(x) = x^2 +3x -10

g(x) = x^2 +3x -10

so how many roots does the system of equations have ?