SOLUTION: Show that the sum of the roots of a quadratic equation is -b/a. Can someone get me started?

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Question 1199134: Show that the sum of the roots of a quadratic equation is -b/a.
Can someone get me started?

Found 4 solutions by josgarithmetic, greenestamps, ikleyn, math_tutor2020:
Answer by josgarithmetic(39617)   (Show Source):
You can put this solution on YOUR website!
Do you know the roots starting from ?

Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!

The quadratic formula says the roots of the quadratic equation are

and

Add the two roots and simplify and see what you get.

Answer by ikleyn(52784)   (Show Source):
You can put this solution on YOUR website!
.

Vieta's formula / (theorem) for quadratic equation.

See and learn from these sources

https://www.andrew.cmu.edu/user/daltizio/Vietas%20Formulas.pdf
https://en.wikipedia.org/wiki/Vieta%27s_formulas#:~:text=In%20mathematics%2C%20Vieta's%20formulas%20relate,%2C%20%22Franciscus%20Vieta%22).

About French mathematician Francois Vieta read from this Wikipedia article

https://en.wikipedia.org/wiki/Fran%C3%A7ois_Vi%C3%A8te

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To see many typical solved problems on Vieta's theorem, look into the lesson
    - Using Vieta's theorem to solve qudratic equations and related problems
in this site.

Also, you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this textbook under the topic "Quadratic equations".

Save the link to this online textbook together with its description

Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson

to your archive and use it when it is needed.

Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!

Method 1

Use the quadratic formula to see the two roots are and

We can simplify things down a bit to say the two roots are and where
When adding those roots together, the "S" terms cancel (since we have a positive S added to a negative S).

We'll then have as the sum of the two roots, where 'a' is nonzero.

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Method 2

The vertex is located at the point (h,k)
The formula for h is

where 'a' is nonzero.
This formula is useful for completing the square to get the equation into vertex form

The equation represents the vertical line through the vertex.
This vertical line is known as the axis of symmetry.

Let the two roots be p and q.
If we knew the roots, we can average them to determine the value of h.
We'll use this fact to isolate .

.... h is the midpoint of p and q

.... plug in h = -b/(2a)

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Method 3

p and q are the roots of

Meaning:

and

Each input (x = p and x = q) lead to an output of y = 0.
i.e. p and q are the x-intercepts on the parabola.

Subtract the equations straight down

From that, either
, or

If the first scenario is the case, then leads to .
This isn't particularly useful.

So we move onto the second scenario.