Questions   
Sort: 
 #14
avatar
0

Hi HumanBeing,

 

I am just confirming Tiggsy's amazing answer (must more efficient than my answer), and that the Guest's comment below his post is incorrect, in fact, n=11/8.

\(\text{Given } \space \space y=ax^2+bx+c \space \space \space \space \space (*) \\ \)

\(\text{If the parabola passes through (1,1) and (4,-7)} \\ \text{Then, they must satisfy (*), thus substituting into the equation yields: } \\ \space\space\space\space\space\space\space\space\space1=a+b+c \space\space\space\space\space\space\space\space\space\space (1) \\ \space\space\space\space-7=16a+4b+c \space\space\space\space(2) \\\)

\( \text{Now, we have a system of two equations but with three unknowns} \\ \text{So, we have to use the last bit of information} \\ \text{Basically, if the greater root is } \space\space\space \sqrt{n} +2 = 2+\sqrt{n} \\ \text{Then, we know that irrational roots come in pairs, I.e. recall the following theorem: } \\ \text{if } \space\space a+\sqrt{b} \space\space\space \text{is a root, then} \space\space\space a-\sqrt{b} \space\space\space \text{must be a root of the polynomial}.\)

\(\text{Hence, } \space\space 2-\sqrt{n} \space\space \text{is a root} \\ \text{Sum of the roots: } \space \space 4 \\ \text{Product of the roots: } \space\space 4-n \\ \text{Apply Vieta's formulae: } -\dfrac{b}{a} = 4 \space\space\space (**) \\ \dfrac{c}{a}=4-n \space\space\space\space (***)\\ \)

Using (**), we can eliminate b from (1) and (2) as follows:
\(b=-4a \\ \text{(1) and (2) becomes: } \space \\ \iff 1=a-4a+c \implies 1=-3a+c\\ \iff -7=16a-16a+c \implies c=-7 \\ \text{Thus, } a=-\dfrac{8}{3}\)

Using (***): 

\(\dfrac{c}{a}=4-n \\ \iff \dfrac{-7}{\dfrac{-8}{3}}=4-n \\ \iff \dfrac{21}{8}=4-n \\ \iff n=1.375= \dfrac{11}{8} \\ \text{Which is the answer}.\)

.
Jul 23, 2023
Jul 22, 2023
 #13
avatar
0
Jul 22, 2023

0 Online Users