If x R then x^2 +2x +a/x^2 +4x+ 3a can take all real values if Answer is a - Maths - Complex Numbers and Quadratic Equations

Question

If x€R then x^2 +2x +a/x^2 +4x+ 3a can take all real values if
? Answer is a€[0,1] Exerts I've already asked this and exert
gave me link to similar questions But pls solve it Pls solve it in
the way that we take denominator to other side and multiply by y
then write the new formed quadratic equation and then take D
>or=0 and again get a quadratic equation and again D>or =0
Pls solve using this method Thanks

Shivam Mishra · Accepted Answer

Dear Student, Please find below the solution to the asked query: We havey=x2+2x+ax2+4x+3a⇒yx2+4xy+3ay=x2+2x+a⇒yx2+4xy+3ay-x2-2x-a=0⇒x2y-1+2x2y-1+3ay-a=0For x to be real,Discriminant≥0⇒22y-12-43ay-ay-1≥0⇒42y-12-43ay-ay-1≥0Dividing both sides by 4, we get,⇒2y-12-3ay-ay-1≥0⇒4y2+1-4y-3ay2-3ay-ay+a≥0⇒4y2+1-4y-3ay2-4ay+a≥0⇒4y2+1-4y-3ay2+4ay-a≥0⇒y24-3a+4ya-1+1-a≥0For above equation to be always true,4-3a>0 and Discriminant≤0Now 4-3a>0⇒a<43Discriminant≤0⇒4a-12-44-3a1-a≤0⇒16a2+1-2a+44-3aa+1≤0Dividing both sides by 4 we get,4a2+1-2a+4-3aa+1≤0⇒4a2+4-8a+4a-3a2-4+3a≤0⇒a2-a≤0⇒aa-1≤0⇒a∈0,1Taking intersection of a<43 and a∈0,1, we get,a∈0,1 Answer Hope this information will clear your doubts about this topic If you have any doubts just ask here on the ask and answer forum and our experts will try to help you out as soon as possible Regards