[ASK] Stuck on a Quadratic Equation

In summary, the given equation with a whole number a has positive roots, and if x_1>x_2, then the difference between x_2 and x_1 can be determined by finding the single value of a, which is 2.
  • #1
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The equation \(\displaystyle (a-1)x^2-4ax+4a+7=0\) with a is a whole number has positive roots. If \(\displaystyle x_1>x_2\) then \(\displaystyle x_2-x_1=...\)
A. –8
B. –5
C. –2
D. 2
E. 8

Since the equation has positive roots then \(\displaystyle x_1>0\) and \(\displaystyle x_2>0\) thus \(\displaystyle x_1+x_2>0\) and \(\displaystyle x_1x_2>0\)

\(\displaystyle x_1+x_2>0\)
\(\displaystyle \frac{-(-4a)}{a-1}>0\)

\(\displaystyle x_1x_2>0\)
\(\displaystyle \frac{4a+7}{a-1}>0\)

However I progressed, I couldn't determine a as a single value and only found it as a set of certain whole numbers. Can you help me to find the single value of a? Once I know that. I guess I can continue on my own.
 
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  • #2
$b^2-4ac = 16a^2 - 4(a-1)(4a+7) > 0 \implies 4(7-3a) > 0 \implies a \in \{0,1,2\}$

of those three possible values for $a$, only $a=2$ yields positive roots ... $x = 3$ & $x = 5$
 

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