SOLUTION: Show that the product of the roots of a quadratic equation is c/a. I see the word product in the application. This tells me to multiply two quadratic formulas and simplify to ge

Question 1207450: Show that the product of the roots of a quadratic equation is c/a.
I see the word product in the application. This tells me to multiply two quadratic formulas and simplify to get c/a.
Yes?
Found 4 solutions by Theo, math_tutor2020, MathTherapy, ikleyn:
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
basically, you show the expression of one root times the other root and then show that it simplifies to c/a.

see my worksheet below.

the distributive law states that (a + b) * (a - b) equals:
a * (a - b) + b * (a - b) which equals:
(a^2 - ab) + (ab - b^2)
which simplifies to:
a^2 - b^2

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

a,b,c are real numbers
'a' is nonzero

The standard template for a quadratic is
ax^2 + bx + c = 0
Divide everything by 'a' to get
x^2 + (b/a)x + c/a = 0
We'll come back to this later.
The key thing to pay attention to is that the leading coefficient is 1.

Let r and s be the two roots of this quadratic.
If x = r is a root then x-r is a factor
Same goes for the other root.
We find that:
(x-r)(x-s) = 0
which rewrites to
x^2-(r+s)x+rs = 0
Here the leading coefficient is 1 as well.

The product of the roots is the "rs" term at the end.
Compare that to the last term in x^2 + (b/a)x + c/a = 0 to see it matches with the c/a.
Both are constant terms not attached to x in any fashion.

Therefore, rs = c/a

For more information, search out "Vieta's Formulas".

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Alternative route

You can use the quadratic formula like tutor Theo has done, but it could get a bit messy.
The good news is that notice how d = b^2 - 4ac is the discriminant so would become

Then that would lead to these roots
and
The numerators multiply to b^2-d when using the difference of squares rule.
Then compute b^2-d = b^2-(b^2-4ac) = 4ac
The denominators multiply to 4a^2

So this is a fairly quick way to arrive at (4ac)/(4a^2) = c/a.

Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!

Show that the product of the roots of a quadratic equation is c/a. I see the word product in the application. This tells me to multiply two quadratic formulas and simplify to get c/a. Yes? No!! It doesn't mean you need to multiply 2 quadratic equations. It merely means that you have to multiply the 2 PARTS (roots) of the quadratic function, as follows: , and so: PRODUCT of the 2 roots: (x₁ * x₂): ------- Multiplying numerators and denominators ---- FOILing numerator

Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.

The short answer to your question is "YES",

although there are much simpler ways to prove this identity.

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