Determining how the given number lies in relation to the roots of a quadratic equationSay we have a quadratic x2+px+q=0If this quadratic has 2 roots, then:For any number, λ, if λ2+pλ+q&lt;0 then λ lies between both roots.and for any number λ, if λ2+pλ+q&gt;0 then if 2λ+p&gt;0, then λ&lt; either of the roots.1) I don't understand graphical (physical) meaning - What is (2λ+p) and what is 2λ+p&lt;0 and 2λ+p&gt;02) Also, what is λ2+pλ+q&gt;0 and λ2+pλ+q&lt;0.

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Determining how the given number lies in relation to the roots of a quadratic equationSay we have a quadratic x2+px+q=0If this quadratic has 2 roots, then:For any number, λ, if λ2+pλ+q&amp;lt;0 then λ lies between both roots.and for any number λ, if λ2+pλ+q&amp;gt;0 then if 2λ+p&amp;gt;0, then λ&amp;lt; either of the roots.1) I don&#039;t understand graphical (physical) meaning - What is (2λ+p) and what is 2λ+p&amp;lt;0 and 2λ+p&amp;gt;02) Also, what is λ2+pλ+q&amp;gt;0 and λ2+pλ+q&amp;lt;0.

Jazlyn Mitchell · Accepted Answer

Since your equation has 2 distinct real roots we can write x2+px+q=(x−α)(x−β) with α&amp;gt;β and α+β=−p.Case 1: Let be λ such that λ2+pλ+q&amp;lt;0. If we use the second writing of the polynomial we have (λ−α)(λ−β)&amp;lt;0 and it is possible if and only if β&amp;lt;λ&amp;lt;α. This means that λ lies between the two roots.Case 2: Let be λ such that λ2+pλ+q&amp;gt;0. Consider 2 subcases:- If 2λ+p&amp;gt;0 then you have: λ&amp;gt;α+β2&amp;gt;β+β2&amp;gt;β. Using the second writing you have now (λ−α)(λ−β)&amp;gt;0 and using the above inequality, the second factor is more than 0. Then also the first factor has to be more than 0. So β&amp;lt;α&amp;lt;λ.- If 2λ+p&amp;lt;0 then you have: λ&amp;lt;α+β2&amp;lt;α+α2&amp;lt;α. Using the second writing you have now (λ−α)(λ−β)&amp;gt;0 and using the above inequality, the first factor is less than 0. Then also the second factor has to be less than 0. So α&amp;gt;β&amp;gt;λ.The meaning of the condition 2λ+p&amp;gt;0 is that λ lies at the right of the x coordinate of the vertex of the parabola. Simmetrically 2λ+p&amp;lt;0 means that λ lies at the left of the x coordinate of the vertex of the parabola.

Determining how the given number lies in relation

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