Question

Help with question 7 part i & ii

Answer 1

(i)
`k\in (-\infty,-16)\cup (0,\infty)`

(ii)
`p\in (9,\infty)`

Explanation:

(i) Given equation

`2x^2+4x+2=k(x+3)`

Rewrite this equation as quadratic equation in the standard form:

`2x^2+4x+2=kx+3k\\ \\2x^2+4x+2-kx-3k=0\\ \\2x^2+(4-k)x+(2-3k)=0`

If the equation has two distinct real roots, then its discriminant is greater than 0. Find the discriminant:

`D=(4-k)^2-4\cdot 2\cdot (2-3k)=16-8k+k^2-16+24k=k^2+16k`

Solve the inequality
`D>0:`

`k^2+16k>0\\ \\k(k+16)>0\Rightarrow k\in (-\infty,-16)\cup (0,\infty)`

(ii) The quadratic expression is always greater than 0 when:

`p>0\\ \\D<0`

Find the discriminant:

`D=6^2-4\cdot p\cdot 1=36-4p`

Solve the inequality
`D<0:`

`36-4p<0\\ \\-4p<-36\\ \\4p>36\\ \\p>9`

Assuming that
`p>0` and
`p>9,` you get

`p>9\Rightarrow p\in (9,\infty)`