Question

Given a quadratic equation px²+px+3q=1+2x has the roots 1/p and qfind the value of p and q

Answer 1

The given equation is:

`px^2+px+3q\text{ = 1+2x}`

We can write it as:

`px^2+px+3q\text{ - 1-2x}=0`

Rearrange the terms, we get:

`px^2-2x+px+(3q-1)=0`

This can be written as:

`px^2+x(p-2)+(3q-1)=0`

Now wrt Standard form of a quadratic equation:

`ax^2+bx+c=0`

we have:

`a\text{ =p}`

`b=p-2`

and

`c\text{ = }3q-1`

We know that product of zeroes :

`q\text{ x }(1)/(p)=(3q-1)/(p)`

then

`3q-1=q`

then

`2q\text{ = 1}`

so that,

`q\text{ =}(1)/(2)`

Sum of roots :

`q+(1)/(p)=(2-p)/(p)`

then

`(qp+1)/(p)=(2-p)/(p)`

then

`qp+1\text{ = 2-p}`

then

`(p)/(2)+p=1`

solving for p, we get:

`p\text{ = }(2)/(3)`

so that, we can conclude that the correct answer is:

`p\text{ = }(2)/(3)`

and

`q\text{ =}(1)/(2)`