Finding the value of a variable in a quadraticThe number of negative integral values of m for which the expression x2+2(m−1)x+m+5 is positive ∀x&gt;1 is?

Question

Finding the value of a variable in a quadraticThe number of negative integral values of m for which the expression x2+2(m−1)x+m+5 is positive ∀x&amp;gt;1 is?

slaastro132z · Accepted Answer

Step 1There are two ways for havingp(x)=x2+2(m−1)x+m+5&amp;gt;0&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;∀x&amp;gt;1Either p(x) has no real root, which means the reduced discriminantΔ′=(m−1)2−(m+5)=(m+1)(m−4)&amp;lt;0is negatve;or p(x) has real roots, say&amp;nbsp;ξ0≤ξ1, i.e.&amp;nbsp;Δ′≥0&amp;nbsp;In this case, the condition means 1 is on the right of the interval of the roots. This is satisfied if and only if&amp;nbsp;p(1)≥0&amp;nbsp;(1 is outside of the interval of the roots or is a root) and, using Vieta&#039;s relations,1&amp;gt;12ξ0+ξ1=1−m

Finding the value of a variable in a

Answered question

Answer & Explanation

New Questions in Algebra I