Quadratic equation

Irene · 2024-05-19 23:17:32

Can some one please help understand the formula. I would really appreciate.

Bob · 2024-05-20 00:06:41

hi Irene,

Sometimes a quadratic equation will factorise

eg.

But often they don't. There's an algebraic 'trick' that allows you to get the solutions when this happens. It is called 'completing the square'.

eg.

This time there is no easy factorisation.

Step 0ne. Shift the number to the other side:

Step Two. Add (half the x coefficient) squared to both sides:

This is guaranteed to make the left hand side a perfect square .... it can be factorised as something^2

Step Three. Factorise the LHS

Step Four. Square root both sides.

Note: Two possible square roots unless that term is zero.

Step Five. Shift that non-x term to the RHS

Many years ago when I was taking my exams we were told we could use that method or use 'the formula' For a long time I did because the formula looked too complicated for me. So you're not alone.

The formula is derived using completing the square, but with letters rather than specific numbers. Here we go

Step One. Divide all by 'a' so the first term is x^2

Note: 0 divided by a is still 0

Step Two. Move the non-x term to the other side.

Step Three. Add (half the x coefficient) squared to both sides:

Step Four. Simplify the RHS a bit.

Step Five. Square root both sides

Step Six. Move the non x term to the other side

Step Seven. Put all over the common denominator and 'loose' the unnecessary + sign

Bob

Phrzby Phil · 2024-05-20 01:53:11

I'll add a bit more to Bob's excellent exposition.

The expression under the radical (the radicand) is called the discriminant.

Note that even before completely solving for the roots, the value (<0, =0, >0) of the discriminant tells us something about the roots and the graph.

=0: both roots are the same, because the formula has ±0. The parabola's vertex touches the x-axis there.
<0: both roots are complex; they are called complex conjugates: a+bi and a-bi. (Any two expressions x+y and x-y are called conjugates.) The parabola does not intersect the x-axis.
>0: both roots are real. The parabola has two distinct x-intersects.

Last edited by Phrzby Phil (2024-05-20 01:55:36)

mathxyz · 2024-05-20 05:46:30

Bob wrote:

hi Irene,
Sometimes a quadratic equation will factorise
eg.
But often they don't. There's an algebraic 'trick' that allows you to get the solutions when this happens. It is called 'completing the square'.
eg.
This time there is no easy factorisation.
Step 0ne. Shift the number to the other side:
Step Two. Add (half the x coefficient) squared to both sides:
This is guaranteed to make the left hand side a perfect square .... it can be factorised as something^2
Step Three. Factorise the LHS
Step Four. Square root both sides.
Note: Two possible square roots unless that term is zero.
Step Five. Shift that non-x term to the RHS
Many years ago when I was taking my exams we were told we could use that method or use 'the formula' For a long time I did because the formula looked too complicated for me. So you're not alone.
The formula is derived using completing the square, but with letters rather than specific numbers. Here we go
Step One. Divide all by 'a' so the first term is x^2
Note: 0 divided by a is still 0
Step Two. Move the non-x term to the other side.
Step Three. Add (half the x coefficient) squared to both sides:
Step Four. Simplify the RHS a bit.
Step Five. Square root both sides
Step Six. Move the non x term to the other side
Step Seven. Put all over the common denominator and 'loose' the unnecessary + sign
Bob

Fabulous study notes and reply.

mathxyz · 2024-05-20 05:47:37

Phrzby Phil wrote:

I'll add a bit more to Bob's excellent exposition.
The expression under the radical (the radicand) is called the discriminant.
Note that even before completely solving for the roots, the value (<0, =0, >0) of the discriminant tells us something about the roots and the graph.
=0: both roots are the same, because the formula has ±0. The parabola's vertex touches the x-axis there.
<0: both roots are complex; they are called complex conjugates: a+bi and a-bi. (Any two expressions x+y and x-y are called conjugates.) The parabola does not intersect the x-axis.
>0: both roots are real. The parabola has two distinct x-intersects.

Good additional notes. Good study notes. I love this formula.

Last edited by mathxyz (2024-05-22 17:26:27)

Irene · 2024-05-22 07:27:44

Thanks.

Irene · 2024-05-23 01:32:07

Thanks all, looks like i would have to spend more time on Algebra.

mathxyz · 2024-05-23 15:52:37

Irene wrote:

Thanks all, looks like i would have to spend more time on Algebra.

Copy and paste this link:

https://m.youtube.com/watch?v=bNxtxiN-FtI&pp=ygUlUHJvZmVzc29yIGxlb25hcmQgcXVhZHJhdGljIGVxdWF0aW9ucw%3D%3D

Last edited by mathxyz (2024-05-23 15:53:06)

Irene · 2024-05-23 19:52:33

Thanks mathxyz. I would do that. I still get confused by completing the square method. But am good with the factorization method.

KerimF · 2024-05-23 23:31:37

Irene wrote:

Thanks mathxyz. I would do that. I still get confused by completing the square method. But am good with the factorization method.

For instance, I personally can't explain how it may be difficult for my brain, sometimes, to understand/solve a new idea/problem (speaking practically), then it gets it very well just by itself after a certain time (after a month, a year or too many years in some cases).
The key point is that I had to be interested to get it very seriously, for one reason or another.

Irene · 2024-05-24 00:38:13

Mathxyz okay and thanks.

mathxyz · 2024-05-24 07:35:09

Irene wrote:

Mathxyz okay and thanks.

Professor Leonard is probably the best math professor online.

Irene · 2024-05-24 09:07:06

How can i get his website??? Or his YouTube channel? Thanks

mathxyz · 2024-05-24 15:59:13

Irene wrote:

How can i get his website??? Or his YouTube channel? Thanks

I don't know his website. However, just type his name using YouTube search. It will take you to his channel.

Irene · 2024-05-24 19:24:45

Tanx

mathxyz · 2024-05-24 21:47:33

Irene wrote:

Tanx

No problem. You're welcome.

Math Is Fun Forum

#1 2024-05-19 23:17:32

Quadratic equation

#2 2024-05-20 00:06:41

Re: Quadratic equation

#3 2024-05-20 01:53:11

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#4 2024-05-20 05:46:30

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#5 2024-05-20 05:47:37

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