For which k are the roots of k(x2+1)=x2+3x–3 real and distinct?
Mathematics
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Question
For which k are the roots of k(x2+1)=x2+3x–3 real and distinct?
1 Answer

1. User Answers calculista
Answer:
The solution for k is the interval (3.5,1.5)
Stepbystep explanation:
we have
[tex]k(x^{2}+1)=x^{2}+3x3[/tex]
[tex]kx^{2}+k=x^{2}+3x3[/tex]
[tex]x^{2}kx^{2}+3x3k=0[/tex]
[tex]}[1k]x^{2}+3x(3+k)=0[/tex]
we know that
If the discriminant is greater than zero . then the quadratic equation has two real and distinct solutions
The discriminant is equal to
[tex]D=b^{2}4ac[/tex]
In this problem we have
a=(1k)
b=3
c=(3+k)
substitute
[tex]D=3^{2}4(1k)(3k)\\ \\D=94(3k+3k+k^{2})\\ \\D=9+12+4k12k4k^{2}\\ \\D=218k4k^{2}[/tex]
so
[tex]218k4k^{2} > 0[/tex]
solve the quadratic equation by graphing
The solution for k is the interval (3.5,1.5)
see the attached figure