Question

Given that α and β are the roots of the quadratic equation
`2x^(2) +6x-7=p`, and α=2β, a) find the value of p. b) form a quadratic equation with roots α+2 and β+2

Answer 1

`\large \boxed{\sf \ \ \ p=-11 \ \ \ }`

Explanation:

Hello,

`\alpha \text{ and } \beta \text{ are the roots of the following equation}`

`2x^2+6x-7=p`

It means that

`2\alpha^2+6\alpha-7=p \\\\2\beta ^2+6\beta -7=p \\\\`

And we know that

`\alpha= 2\cdot \beta`

So we got two equations

`2(2\beta)^2+6\cdot 2 \cdot \beta -7=p \\\\<=>8\beta^2+12\beta -7=p\\\\ and \ 2\beta ^2+6\beta -7=p \ So \\\\\\8\beta^2+12\beta -7 = 2\beta ^2+6\beta -7\\\\<=>6\beta^2+6\beta =0\\\\<=>\beta(\beta+1)=0\\\\<=> \beta =0 \ or \ \beta=-1`

For
`\beta =0, \ \ \alpha =0, \ \ p = -7`

For
`\beta =-1, \ \ \alpha =-2, \ \ p= 2-6-7=-11, \ p=2*4-12-7=-11`

I assume that we are after two different roots so the solution for p is p=-11

b)
`\alpha +2 =-2+2=0 \ and \ \beta+2=-1+2=1`

So a quadratic equation with the expected roots is

`x(x-1)=x^2-x`

Hope this helps.

Do not hesitate if you need further explanation.

Thank you