Solve for x^2_1 + x^2_2 + y^2_1 + y^2_2 in y=mx+1 & y=x^4+2x^2+x

[juantheron]

If line \(\displaystyle y=mx+1\) intersect the curve \(\displaystyle y=x^4+2x^2+x\) at two distinct points

Then value of \(\displaystyle x^2_{1}+x^2_{2}+y^2_{1}+y^2_{2} = \)

My Trail:: Solving these two equation, $x^4+2x^2+x=mx+1\Rightarrow x^4+2x^2+(1-m)x-1=0$

Now how can i calculate \(\displaystyle m\) and also find value of that expression,

Thanks

 

[jvicens]

for your question! To find the value of m, you can use the quadratic formula to solve the equation $x^4+2x^2+(1-m)x-1=0$ for x. Once you have the values of x, you can substitute them into the equation y=mx+1 to find the corresponding y values. Then, you can plug those values into the expression $x^2_{1}+x^2_{2}+y^2_{1}+y^2_{2}$ to find the sum of the squares of the two x and two y values. I hope this helps!