Answer:
[tex]\large \boxed{\sf \ \ \ p=-11 \ \ \ }[/tex]
Step-by-step explanation:
Hello,
[tex]\alpha \text{ and } \beta \text{ are the roots of the following equation}[/tex]
[tex]2x^2+6x-7=p[/tex]
It means that
[tex]2\alpha^2+6\alpha-7=p \\\\2\beta ^2+6\beta -7=p \\\\[/tex]
And we know that
[tex]\alpha= 2\cdot \beta[/tex]
So we got two equations
[tex]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[/tex]
For [tex]\beta =0, \ \ \alpha =0, \ \ p = -7[/tex]
For [tex]\beta =-1, \ \ \alpha =-2, \ \ p= 2-6-7=-11, \ p=2*4-12-7=-11[/tex]
I assume that we are after two different roots so the solution for p is p=-11
b) [tex]\alpha +2 =-2+2=0 \ and \ \beta+2=-1+2=1[/tex]
So a quadratic equation with the expected roots is
[tex]x(x-1)=x^2-x[/tex]
Hope this helps.
Do not hesitate if you need further explanation.
Thank you
