The two solutions of the equation $x^2+bx+18=0$ are in the ratio of $2$ to $1$ for some values of $b.$ What is the largest possible value of $b?$
I notice that the equation is a quadratic, and I wonder if I can use the quadratic formula or factor it somehow. I think about what it means for the solutions to be in the ratio of $2$ to $1.$ That means that if one solution is $x_1,$ the other one is $2x_1.$ So the solutions are not equal, and the equation has two distinct roots. I recall that the sum of the roots of a quadratic equation is equal to the opposite of the coefficient of the linear term, divided by the leading coefficient. That is, $x_1+2x_1=-b/1.$ Simplifying, I get $3x_1=-b.$ I also recall that the product of the roots of a quadratic equation is equal to the constant term, divided by the leading coefficient. That is, $x_1\cdot 2x_1=18/1.$ Simplifying, I get $2x_1^2=18.$ I can solve for $x_1$ by dividing both sides by $2$ and taking the square root. I get $x_1=\pm 3.$ Since $x_1$ can be either positive or negative, I have two possible cases to consider. If $x_1=3,$ then $2x_1=6$ and $3x_1=-b.$ So $b=-9.$ If $x_1=-3,$ then $2x_1=-6$ and $3x_1=-b.$ So $b=9.$ The largest possible value of $b$ is $9.$ 