sum and product of roots of X^2 + 4X + 10 = 0

[lewysangel]

How do you find each of these? Any help would be appreciated. Here is an example problem.

X^2 + 4X + 10 = 0

We need to roots, the sum of the roots and the product of the roots.

 

[galactus]

Re: sum and product of roots

There's an easy way besides finding the roots and adding them up or multiplying them.

For the sum of the roots: \(\displaystyle \frac{-b}{a}\)

For the product of the roots: \(\displaystyle \frac{c}{a}\)

 

[pka]

Re: sum and product of roots

\(\displaystyle \left( {x - r_1 } \right)\left( {x - r_2 } \right) = x^2 - \left( {r_1 + r_2 } \right)x + \left( {r_1 r_2 } \right)\)

 

[soroban]

Re: sum and product of roots

Hello, lewysangel!

How do you find each of these? Any help would be appreciated. Here is an example problem.

X^2 + 4X + 10 = 0

We need to roots, the sum of the roots and the product of the roots.

Your question is not clear . . .

I would guess that we are to find the two roots of the quadratic equation.
. . then find the sum of the roots and the product of the roots.

If this is true, what's stopping you?


Using the Quadratic Formula:

. . \(\displaystyle x \;=\;\frac{-4 \pm\sqrt{4^2 - 4(1)(10)}}{2(1)} \;=\;\frac{-4\pm\sqrt{-24}}{2} \;=\;\frac{-4\pm2\sqrt{6}i}{2} \;=\;-2 \pm \sqrt{6}i\)

\(\displaystyle \text{The two roots are: }\;x\;=\;-2 + \sqrt{6}i,\;-2 - \sqrt{6}i\)


\(\displaystyle \text{Their sum is: }\;(-2 + \sqrt{6}i) + (-2 - \sqrt{6}i) \;=\;\boxed{-4}\)

\(\displaystyle \text{Their product is }\;(-2 + \sqrt{6}i)(-2 - \sqrt{6}1) \;=\;4 - 6i^2 \;=\;4 + 6 \;=\;\boxed{10}\)