And why (x−α)(x−β) where α and β are its two roots?And why is the sum of roots i.e. α+β=−ba and product of roots αβ=ca?

Step 1Let an arbitrary quadratic expression be:q(x)=ax2+bx+cDefineα=12a(−b+b2−4ac)β=12a(−b−b2−4ac)Then a little algebra shows thatα+β=−baαβ=caa(x−α)(x−β)=ax2−a(α+β)x+aαβ=ax2+bx+cYou should now be able to see the answer to all of your questions.

Step by step solution to "And why (x−α)(x−β) where α and β are..."

High School
Algebra
Algebra I
Quadratic function and equation

chechemuaen7

Answered question

2022-03-16

And why

(x - α) (x - β)

where

α

and

β

are its two roots?
And why is the sum of roots i.e.

α + β = - \frac{b}{a}

and product of roots

α β = \frac{c}{a}

Answer & Explanation

presistlad

Beginner2022-03-17Added 4 answers

Step 1
Let an arbitrary quadratic expression be:

q (x) = a x^{2} + b x + c

Define

α = \frac{1}{2 a} (- b + \sqrt{b^{2} - 4 a c})

β = \frac{1}{2 a} (- b - \sqrt{b^{2} - 4 a c})

Then a little algebra shows that

α + β = - \frac{b}{a}

α β = \frac{c}{a}

a (x - α) (x - β) = a x^{2} - a (α + β) x + a α β = a x^{2} + b x + c

You should now be able to see the answer to all of your questions.

diesel817637dsf

Beginner2022-03-18Added 13 answers

Step 1
Without using quadratic formula, we can do as follows.
If,

a \neq 0

and

α, β

are the roots of

a x^{2} + b x + c

then,

a x^{2} + b x + c = a (x - α) (x - β)

Step 2
We have,

a x^{2} + b x + c = a (x - α) (x - β)

a x^{2} + b x + c = a x^{2} - a (α + β) x + a α β

b = - a (α + β), c = a α β

α + β = \dots α β = \dots

I leave the rest of the last line to you.

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