Question

The equation px²+px+3q=1+2x has roots 1/p and q

(a) Find the values of p and of q​

Answer 1

p = 2/3

q = 1/2

Explanation:

The given equation is ,

`\sf\to px^2 + px + 3q = 1 + 2x`

We can write it as ,

`\sf\to px^2 + px + 3q - 1 -2x=0`

Rearrange the terms ,

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

This can be written as ,

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

Now wrt Standard form of a quadratic equation ,

`\bf \implies ax^2+bx + c = 0`

we have ,

• a = p
• b = p - 2
• c = 3q - 1

We know that product of zeroes :-

`\to \sf q * (1)/(p) = (3q-1)/(p ) \\\\\sf\to 3q - 1 = q \\\\\sf\to 2q = 1 \\\\\sf\to \boxed{ q =(1)/(2)}`

Sum of roots :-

`\to \sf q + (1)/(p) = (2-p)/(p) \\\\\sf\to ( qp + 1)/(p)= (2-p)/(p) \\\\\sf\to qp + 1 = 2 - p \\\\\sf\to p/2 + p = 1 \\\\\sf\to 3p/2 = 1 \\\\\sf\to \boxed{ p =(2)/(3)}`