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Using the formula to slove the given quadratic to find the roots there is a term involving the square-root of (-1)2 - 4(-40)(k). This expression has to be positive or zero if the roots are real. So 1+160k has to be positive or zero (ie >=0). So k>=-1/160
If you have a quadratic function with real coefficients then it can have: two distinct real roots, or a real double root (two coincidental roots), or no real roots. In the last case, it has two complex roots which are conjugates of one another.
That depends on the value of its discriminant if its less than zero then it has no real roots.
The equation ax^2 + bx + c = 0 where a, b and c are real and a is non-zero has discriminant D = b^2 – 4ac. Then,if D > 0 the equation has two real roots which are distinct;if D = 0 the equation has two real roots which are coincident;if D < 0 the equation has two roots which form a complex conjugate pair (advanced mathematics only).
There is no difference between real solutions and real roots.
Child stop trying to cheat on your homework!