Question

Given that α and β are the roots of the quadratic equation Formula1" title="2x^{2} +6x-7=p" alt="2x^{2} +6x-7=p" align="absmiddle" class="latex-formula">, and α=2β, a) find the value of p. b) form a quadratic equation with roots α+2 and β+2

Answer 1

Answer:

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

Step-by-step 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