Solving a Quadratic Equation: A Puzzling Problem

[Mastiff_Lover]

Hi! I have this Math 11 problem. I do my math through Distance Ed. and there are no examples like this one in the supplied lessons. If anyone could help me with this question, that would be great!

Homework Statement

Find the equation of the quadratic function with a vertex at (1,3) passing through the points (2,1). Write your answer in the form y=ax^2+bx+c.

The Attempt at a Solution

Using these two points (one being the vertex), I am able to write an equation, but in the form of y=a(x-h)+k. Below is the answer I get for that:
y=-2(x-1)+3
However, I was wondering if I can just convert this answer to the other quadratic form, or do I have to do the question an entirely different way.
Any comments/help on ways to do this question appreciated.
Thanks,
Mastiff_Lover

 

[rock.freak667]

Well you did the question in the simplest way. But on point, you can just expand it to get it in the for y=ax²+bx+c

 

[Mastiff_Lover]

Hi! Thanks for the reply! However, how do I expand it to get it for y=ax^2+bx+c?

 

[rock.freak667]

Using these two points (one being the vertex), I am able to write an equation, but in the form of y=a(x-h)²+k. Below is the answer I get for that:
y=-2(x-1)²+3

Note the corrected thing in red.

Hi! Thanks for the reply! However, how do I expand it to get it for y=ax^2+bx+c?

Expanding brackets is a fundamental part in algebra so you should learn this well.

(a+b)(c+d)=a(c+d)+b(c+d) = ac+ad + bc+bd

in your case you have (x-1)²=(x-1)(x-1)

EDIT: and well you should know k(a+b) = ka+kb

 

[Mastiff_Lover]

Thanks!Give me a minute to think this out

 

[Mastiff_Lover]

Hi! Ok! I think I've got the answer. Here it is below:
y=-2(x-1)^2+3
y=(2x+1)(x-1)+3
y=-2x^2+2x+x-1+3
y=-2x^2+3x+2
Do you think this would be the answer?
Thanks,
Mastiff_Lover

 

[rock.freak667]

Hi! Ok! I think I've got the answer. Here it is below:
y=-2(x-1)^2+3
y=(2x+1)(x-1)+3
y=-2x^2+2x+x-1+3
y=-2x^2+3x+2
Do you think this would be the answer?
Thanks,
Mastiff_Lover

Try expanding (x-1)² and then multiply that by -2

 

[Mastiff_Lover]

Ok! I just thought you could do it both ways, but I guess not! So, here it is the way you said. Is this correct?
y=-2(x-1)(x-1)+3
y=-2(x^2-x-x+1)
y=-2(x^2-2x+1)
y=2x^2-4x+2+3
y=2x^2-4x+5
Thanks,
Mastiff_Lover

 

[rock.freak667]

Ok! I just thought you could do it both ways, but I guess not! So, here it is the way you said. Is this correct?
y=-2(x-1)(x-1)+3
y=-2(x^2-x-x+1)
y=-2(x^2-2x+1)
y=2x^2-4x+2+3
y=2x^2-4x+5[/B
Thanks,
Mastiff_Lover

you multiplied by 2 and not -2 in this line

 

[Mastiff_Lover]

Oh! I am so sorry! It is late here and I am a bit tired. I tend to make these silly mistakes when I am tired. Anyways, hopefully this is the correct answer:
y=-2x^2+4x+1
Thanks,
Mastiff_Lover

 

[rock.freak667]

Yes that should be correct now.

 

[Mastiff_Lover]

OK! Thanks again for the help!

 

[Mark44]

Ok! I just thought you could do it both ways, but I guess not! So, here it is the way you said. Is this correct?
y=-2(x-1)(x-1)+3
y=-2(x^2-x-x+1)
y=-2(x^2-2x+1)
y=2x^2-4x+2+3
y=2x^2-4x+5
Thanks,
Mastiff_Lover

Rock.freak pointed out one error you made. The error actually started at an earlier line where you forgot to bring along the 3 term. Also, it is not incorrect to rewrite the equation on each line, but you can use a chain of equal signs to indicate that you have a chain of equal expressions. If you're applying some operation to both sides of an equation, you should definitely write both sides in each step, but if all you're doing is simplifying an expression on one side of an equation, you can do it like so:

y=-2(x-1)(x-1)+3 = -2(x² -2x + 1) + 3
= -2x² + 4x -2 + 3 = -2x² + 4x + 1

So y = -2x² + 4x + 1

Since you are in 11th grade algebra (and so presumably have had a previous course in this subject), you should be very comfortable working with the squares of binomials such as (x + a)², which equals x² + 2ax + a². You should be able to do simple binomials like this without having to do a lot of steps. If this is not something you are very familiar with, I would suggest you review binomial multiplication in whatever text you are using and work enough problems that it becomes easy.

 

[Mastiff_Lover]

Hi! Thanks for the tips. I am pretty comfortable in binomial multiplication. However, I am going to do a little bit more practise. I think because I was tired, I was making a few silly mistakes.

Rock.freak pointed out one error you made. The error actually started at an earlier line where you forgot to bring along the 3 term. Also, it is not incorrect to rewrite the equation on each line, but you can use a chain of equal signs to indicate that you have a chain of equal expressions. If you're applying some operation to both sides of an equation, you should definitely write both sides in each step, but if all you're doing is simplifying an expression on one side of an equation, you can do it like so:

y=-2(x-1)(x-1)+3 = -2(x² -2x + 1) + 3
= -2x² + 4x -2 + 3 = -2x² + 4x + 1

So y = -2x² + 4x + 1

Since you are in 11th grade algebra (and so presumably have had a previous course in this subject), you should be very comfortable working with the squares of binomials such as (x + a)², which equals x² + 2ax + a². You should be able to do simple binomials like this without having to do a lot of steps. If this is not something you are very familiar with, I would suggest you review binomial multiplication in whatever text you are using and work enough problems that it becomes easy.

 

[paulatoot]

Hi! I have this Math 11 problem. I do my math through Distance Ed. and there are no examples like this one in the supplied lessons. If anyone could help me with this question, that would be great!

Homework Statement

Find the equation of the quadratic function with a vertex at (1,3) passing through the points (2,1). Write your answer in the form y=ax^2+bx+c.

The Attempt at a Solution

Using these two points (one being the vertex), I am able to write an equation, but in the form of y=a(x-h)+k. Below is the answer I get for that:
y=-2(x-1)+3
However, I was wondering if I can just convert this answer to the other quadratic form, or do I have to do the question an entirely different way.
Any comments/help on ways to do this question appreciated.
Thanks,
Mastiff_Lover

where do you got the -2 as the a?

 

[eumyang]

where do you got the -2 as the a?

Take the vertex form,
y = a(x - h)² + k
and plug in the vertex for h and k. Now, knowing that this parabola passes through the point (2, 1), what do you think we should do next?