Question 348125
In the general form of the quadratic equaiton, {{{ax^2 + bx + c = 0}}}, the value of the discriminant, {{{b^2 - 4ac}}}, will indicate the number of real solutions:<ul><li>{{{b^2 - 4ac = 0}}} means one real solution.</li><li>{{{b^2 - 4ac > 0}}} means two real solutions.</li><li>{{{b^2 - 4ac < 0}}} means no real solutions.</li></ul>
Your discriminant is:
{{{k^2 - 4(1)(2) = k^2 - 8}}}
So if {{{k^2 - 8 = 0}}} you get one real solution. Solving this for k will tell us the k's that will result in one real solution. Adding 8 to each side we get:
{{{k^2 = 8}}}
This gives us
{{{k = sqrt(8) = 2sqrt(2)}}} or {{{k = -2sqrt(2)}}}
For either of these values of k, there will be one real solution.<br>
For two real solutions we want {{{k^2 - 8 > 0}}}. This means we need {{{abs(k) > 2sqrt(2)}}}. In other words, {{{k > sqrt(2)}}} or {{{k < -sqrt(2)}}}.<br>
For no real solutions we want {{{k^2 - 8 < 0}}}. This means we need {{{abs(k) < 2sqrt(2)}}}. In other words, {{{k < sqrt(2)}}} and {{{k > -sqrt(2)}}}.