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

Solution:

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 :

$q\text{ x }\frac{1}{p}=\frac{3q-1}{p}$

then

$3q-1=q$

then

$2q\text{ = 1}$

so that,

$q\text{ =}\frac{1}{2}$

Sum of roots :

$q+\frac{1}{p}=\frac{2-p}{p}$

then

$\frac{qp+1}{p}=\frac{2-p}{p}$

then

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

then

$\frac{p}{2}+p=1$

solving for p, we get:

$p\text{ = }\frac{2}{3}$

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

$p\text{ = }\frac{2}{3}$

and

$q\text{ =}\frac{1}{2}$