Question

If (ax+2)(bx+7)=15x2+cx+14 for all values of x, and a+b=8, what are the 2 possible values fo c

Answer 1

Given:

`(ax+2)(bx+7)=15x^2+cx+14`

And

`a+b=8`

Required:

To find the two possible values of c.

Step-by-step explanation:

Consider

`\begin{gathered} (ax+2)(bx+7)=15x^2+cx+14 \\ abx^2+7ax+2bx+14=15x^2+cx+14 \end{gathered}`

So

`\begin{gathered} ab=15-----(1) \\ 7a+2b=c \end{gathered}`

And also given

`a+b=8---(2)`

Now from (1) and (2), we get

`\begin{gathered} a+(15)/(a)=8 \\ \\ a^2+15=8a \\ \\ a^2-8a+15=0 \end{gathered}`
`a=3,5`

Now put a in (1) we get

`\begin{gathered} (3)b=15 \\ b=(15)/(3) \\ b=5 \\ OR \\ b=(15)/(5) \\ b=3 \end{gathered}`

We can interpret that either of a or b are equal to 3 or 5.

When a=3 and b=5, we have

`\begin{gathered} c=7(3)+2(5) \\ =21+10 \\ =31 \end{gathered}`

When a=5 and b=3, we have

`\begin{gathered} c=7(5)+2(3) \\ =35+6 \\ =41 \end{gathered}`

Final Answer:

The option D is correct.

31 and 41