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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.

 Jun 24, 2020
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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. 

 Jun 24, 2020
edited by thelizzybeth  Jun 24, 2020

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