The way this is written, you cannot use -1/2 or -2. Is that correct? Both of them are minus. So the only things you have going for you is 0 and 5
x^2 > x + 1 when x = 5
5^2 > 5 + 1 and his conjecture is true.
25 > 6
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What about 0?
0^2 = 0 so
0 > 0 + 1 and the conjecture is false.
Answer:
5
Step-by-step explanation:
Given inequality,
[tex]x^2 > x + 1[/tex]
For [tex]x=-\frac{1}{2}[/tex]
[tex](-\frac{1}{2})^2 > -\frac{1}{2} + 1[/tex]
[tex]\implies \frac{1}{4} > \frac{1}{2}[/tex] ( False )
For [tex]x=-2[/tex]
[tex](-2)^2 > -2 + 1[/tex]
[tex]\implies 4 > 1[/tex] ( true )
For [tex]x=0[/tex]
[tex](0)^2 > 0+ 1[/tex]
[tex]\implies 0 > 1[/tex] ( False )
For [tex]x=5[/tex]
[tex](5)^2 > 5 + 1[/tex]
[tex]\implies 25 > 6 [/tex] ( True )
Thus, x = -2 and 5 are the solutions of the given inequality,
But, negative numbers are not allowed.
Hence, x = 5 is a counterexample to kelvins conjecture.
