Answer:
The values of 'k' are -9 or 9
Step-by-step explanation:
Given:
The quadratic equation is given as:
[tex]x^2+kx+18=0[/tex]
The roots of the equation are [tex]\alpha\ and\ 2\alpha[/tex].
Now, we know that, for a quadratic equation of the form [tex]ax^2+bx+c=0[/tex]
The sum of roots = [tex]-\frac{b}{a}[/tex]
Product of roots = [tex]\frac{c}{a}[/tex]
Now, for the given quadratic equation, we have:
[tex]a=1,b=k,c=18[/tex]
Therefore, using the sum of roots, we get:
[tex]\alpha+2\alpha =-\frac{k}{1}\\\\3\alpha =-k\\\\k=-3\alpha ---1[/tex]
Now, using product of roots, we get:
[tex]\alpha \times 2\alpha =\frac{18}{1}\\\\2\alpha^2=18\\\\\alpha^2=9\\\\\alpha=\pm\sqrt9=\pm3[/tex]
Now, plug in the values of [tex]\alpha[/tex] in equation (1). This gives,
[tex]k=-3(3)=-9\ or\ k=-3(-3)=9\\\\\therefore k=\pm9[/tex]
Hence, the values of 'k' are -9 or 9
