Question

Let u and v be the solutions to 3x^2 + 5x + 7 = 0. Find (u/v) + (v/u)

Answer 1

=-0.809

Explanation:

3x^2+5x+7=0 complete the square

3x^2+5x=-7 divide both sides by 3

x^2+5/3 x = -7/3 to complete the square add term(b/2)^2=((5/3)/2)²=25/36

X^+5/3 x+25/36=-7/3 +25/36 factorize

(x+5/6)²= -59/36

x+5/6= + or - √59/36

solution for x= -5/6-√59/6i (u) OR -5/6+√59/6i (v)

u/v + v/u=(-5/6-√59/6i)/(-5/6+√59/6i) +(-5/6+√59/6i)/(-5/6-√59/6i)=-17/21

=-0.809

Answer 2

`\large \boxed{\sf \ \ \ -(17)/(21)=-0.80952... \ \ \ }`

Explanation:

Hello,

First of all we can verify that 0 is not a solution of the equation so that we can divide by u or v

`(v)/(u)+(u)/(v)=(u^2+v^2)/(uv)=((u+v)^2-2uv)/(uv)=((u+v)^2)/(uv)-2`

And we know that

`u+v=-(5)/(3)\\\\uv=(7)/(3)\\\\\\as \ \ (x-u)(x-v)=x^2-(u+v)x+uv`

So it comes

`(v)/(u)+(u)/(v)=(5^2*3)/(7*3^2)-2=(25-42)/(21)=-(17)/(21)=-0.80952...`

Hope this helps.

Do not hesitate if you need further explanation.

Thank you