If h is an integer, find the greatest value of h such that x^{2} + (h+2)x + (h+5) = 0 has no real roots.
+310 If a quadratic has no real roots, the discriminant \(b^2-4ac\), where a=1, h+2=b and h+5=c, is less than 0.
So let's plug in our values.
\((h+2)^2-4(h+5)<0\).
Expand:
\(h^2+4h+4-4h-20<0\)
Simplify by combining like terms:
\(h^2-16<0\)
This can be factorized into:
\((h+4)(h-4)<0\)
This can only happen when one is positive, and one term is negative. h+4 is obviously greater than h-4, so h+4 must be the positive one.
\(h+4>0\) and \(h-4<0\)
\(h>-4\) and \(h<4\).
Combine to get \(-4 < h < 4\)
Since h is the greatest positive integer, the answer is 3.
