Question

If α and β are the zeroes of the given polynomial , and α+ β= αβ then the value of p is px2-2x+3p

Answer 1

Question:

If α and β are the zeroes of the given polynomial , and α+ β= αβ then the value of p is Px²-2x+3p.

To find:

★ To find the value of p .

Answer:

$The \: value \: of \: \underline{ \:\underline{ \: \bold{ \sf \pink{p} } \: is \: \bf{ \sf \pink{ \frac{1}{6}} } \: }\: }$

Given:

★ An equation is given,

$\sf P {x}^{2} - 2x + 3p = 0$

★ And also given that,$\alpha + \beta = \alpha \beta$

Step-by-step explanation:

$P {x}^{2} - 2x + 3p = 0$

It's of the form,$a {x}^{2} + bx + c = 0$

Where,$a=1,b=\: -2,c=3p$

$\boxed{Sum \: of \: roots = \frac{ - a}{b} }$

$\mapsto \alpha + \beta = \frac{ - 1}{ - 2}$

$\mapsto \boxed{\alpha + \beta = \frac{ 1}{ 2} }$

$\boxed{Product \: of \: roots = \frac{ c}{a} }$

$\mapsto \alpha \beta = \frac{ 3p}{1}$

$\mapsto \boxed{\alpha \beta = 3p}$

$\underline{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: }$

Given,

$\implies \alpha + \beta = \alpha \beta$

Now substituting the values,

$\implies \frac{1}{2} = 3p$

$\implies 6p = 1$

$\implies \boxed{p = \frac{1}{6} }$

$\therefore The \: value \: of \: \underline{ \: \bold{p } \: is \: \bf{ \frac{1}{6} } \: }$

Formula used:

$\bigstar Sum \: of \: roots = \frac{ - a}{b}$

$\bigstar Product \: of \: roots = \frac{ c}{a}$

More information:

Formation of quadratic equation

( if the roots α and β are given )

${x}^{2} -x ( \alpha + \beta ) + \alpha \beta = 0$

Answer 2

Answer:

-6 =P

Step-by-step explanation:

The value of p is -6 .