For one real solution: k=6 or k=-6 For two real solutions: k>6 or k<-6
Practice makes perfect
Recall that we can tell the number of real solutions a quadratic equation has by calculating its discriminant. For a quadratic equation of the form a^2+bx+c=0, the discriminant is given by the expression shown below.
b^2-4ac
We can find the different possibilities corresponding to different discriminant values in the following table.
Discriminant
Number of real solutions
b^2-4ac >0
The function intersects the x-axis two times. There are two real solutions.
b^2-4ac = 0
The function intersects the x-axis one time. There is one real solution.
b^2-4ac < 0
The function does not intersect the x-axis. There are no real solutions.
Comparing the expression 1*x^2+kx+9 with ax^2+bx+c, we can identify that for this case a=1, b=k and c=9. For the equation to have just one real solution the discriminant must vanish. Thus, we have the condition b^2-4ac =0. Let's see its implications for the possible values for k.
Notice that since k is squared the inequality is satisfied as long as k>6 or k<-6. Hence, for this values of k the equation would have two real solutions.
