If one root of the equation x2+px+q=0 is 3 times the other prove that 3p2=16q.

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Filo · Accepted Answer

We know that if m and n are the roots of a quadratic equation ax2+bx+c=0, the sum of the roots is m+n=−ab​ and the product of the roots is mn=ac​.Let m and n be the roots of the given quadratic equation x2+px+q=0. It is given that one of the root is three times the other, therefore,m=3n........(1)The equation x2+px+q=0 is in the form ax2+bx+c=0 where a=1,b=p and c=q.  Using equation 1, the sum of the roots is:m+n=−ab​=−1p​=−p⇒m+n=−p⇒3n+n=−p⇒4n=−p⇒n=−4p​....(2)Using equation 1, the product of the roots is ac​ that is:mn=ac​=1q​=q⇒n=−p⇒(3n×n)=q⇒3n2=q......(3) Now, substitute equation 2 in equation 3 as follows: (3n×n)=q⇒3(−4p​ )2=q⇒(3×16p2​ )=q⇒163p2​=q⇒3p2=16q  Hence, the value of 3p2=16q.

Question Text	If one root of the equation $x^{2} + p x + q = 0$ is 3 times the other prove that $3 p^{2} = 16 q$ .
Answer Type	Text solution:1
Upvotes	150

If one root of the equation $x^{2} + p x + q = 0$ is 3 times the other prove that $3 p^{2} = 16 q$ .

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