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If one root of the equation x^2 +px + q = 0 is 3 times the oth

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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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If one root of the equation

x^{2} + p x + q = 0

is 3 times the other prove that

3 p^{2} = 16 q

8 mins ago

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Text solutionVerified

We know that if

m

and

n

are the roots of a quadratic equation

a x^{2} + b x + c = 0

, the sum of the roots is

m + n = - \frac{b}{a}

and the product of the roots is

mn = \frac{c}{a}

Let

m

and

n

be the roots of the given quadratic equation

x^{2} + p x + q = 0

. It is given that one of the root is three times the other, therefore,

m = 3 n ........ (1)

The equation

x^{2} + p x + q = 0

is in the form

a x^{2} + b x + c = 0

where

a = 1, b = p

and

c = q

Using equation 1, the sum of the roots is:

m + n = - \frac{b}{a} = - \frac{p}{1} = - p \Rightarrow m + n = - p \Rightarrow 3 n + n = - p \Rightarrow 4 n = - p \Rightarrow n = - \frac{p}{4} .... (2)

Using equation 1, the product of the roots is

\frac{c}{a}

that is:

mn = \frac{c}{a} = \frac{q}{1} = q

\Rightarrow n = - p \Rightarrow (3 n \times n) = q \Rightarrow 3 n^{2} = q ...... (3)

Now, substitute equation 2 in equation 3 as follows:

(3 n \times n) = q \Rightarrow 3 (- \frac{p}{4})^{2} = q \Rightarrow (3 \times \frac{p ^{2}}{16}) = q \Rightarrow \frac{3 p ^{2}}{16} = q \Rightarrow 3 p^{2} = 16 q

Hence, the value of

3 p^{2} = 16 q

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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 x2+px+q=0 is 3 times the other prove that 3p2=16q.

Text solutionVerified

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