Quadratic formula is pretty useless here
Given that x 2 + 5 x + 6 is a prime number, determine the smallest integer value of x .
Details and Assumptions :
- If you think no such solution exists, input 9 9 9 as your answer.
The answer is -4.
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13 solutions
My analysis: x^^2 is odd when x is odd, even when x is even. Same for 5x. Thus the sum of the terms is always even, which means the prime number is 2. Then the light turned on - x must be negative.
By inspection, -1 solves the equation. Also -1 is "SMALLER" than -4 (as in a bank overdraft, for instance). -4 is "LESS THAN" -1 by the commonly-accepted definition of the relation.
Parlor-trick framing of the question, which is a regular occurrence here on "brilliant".org
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This isn't a parlor trick - it is a maths question, and it is common for the term "smallest value" to mean the value which is the least amongst all possible values. There is no planet in which -1 would be considered smaller than -4. Even in a bank overdraft if my balance is -£4 I have less money then I would if my balance is -£1, as in the first scenario I owe the bank more money (£4 as opposed to £1); and if I own more I have less in total.
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"There is no planet" is an unhelpful overstatement. "Small" is frequently used in contexts like "1+x is a good approximation for exp(x), when x is small". It is well-understood that "small" in this context means "sufficiently close to 0", so absolutely -1 can be considered smaller than -4.
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@Erick Wong – I have never heard anyone claim that -1 is smaller than -4, unless they have only just been introduced to negative numbers and are still confused. I still think Richard claiming the answer is a Parlor Trick is disingenuous at best - the question was fine, just that Richard misunderstood it.
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@Tony Flury – It is a parlor trick, because the question is left deliberately ambiguous. Despite your obtuse claims to the contrary, you have just heard two people independently claim that -1 is smaller than -4. Would you disagree that -10^100 could reasonably be called a "large negative number", or would you insist on calling it small? The truth is that both interpretations of "smaller" are common, so one is left to guess which interpretation is intended. This is simply a poorly-written question.
how is that if its a prime number
then, ∣ x + 2 ∣ = 1 or ∣ x + 3 ∣ = 1 ,
can you please explain.
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Prime numbers are those which are only divisible by 1 and themselves. We are looking for cases where x 2 + 5 x + 6 is prime, which is equivalent to looking for prime numbers ( x + 2 ) ( x + 3 ) . Or in other words, we are looking for prime numbers which factor as ( x + 2 ) ( x + 3 ) . This seems to be a contradiction to the definition of prime numbers, but it is not; if one of the two numbers is equal to 1 and the other is a prime number, there is no contradiction. But there is still something we are overlooking:
If x < − 3 , then x + 3 < 0 and x + 2 < 0 . Also the product of two negative numbers is positive. Therefore there is one more possible resolution to the contradiction above: the case where one of the two numbers is − 1 and the other is the negative of a prime number.
Putting these two cases together into one compact statement gives ∣ x + 2 ∣ = 1 or ∣ x + 3 ∣ = 1 .
Because a prime number does not have any other factors other than itself and 1 .
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That is not a mathematical prove. Can you explain why only the modulus of the factors is equal to 1
Is -4 a smaller integer than -1?
x = -1 also results in a prime (2), and satisfies the above composite check. since |-1 + 2| = |1| = 1
I have another reasoning. For all even and odd integers x, x^2+5x+6 will be an even number. Given that x^2+5x+6 is prime and that the only even prime number is 2, x^2+5x+6=2. The roots of the quadratic equations are -1 and -4. The smallest of them is -4 and there is your answer.
if value of x determined by formula , -b +Squreroot b^2-4ac the of x comes 1 or -6 .
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That formula only works if you are trying to make the formula to be zero - which is not the question being asked.
Why isn't it -1?
There are infinitely many values so how do we know that -4 is correct?
The smallest prime number is 2, so we can try with x 2 + 5 x + 6 = 2 x 2 + 5 x + 4 = 0 ( x + 1 ) ( x + 4 ) = 0 So an integer wich makes x 2 + 5 x + 6 be a prime number is x = − 4 . Also we can realize that for any other prime number, the factorization ( x + a ) ( x + b ) will not exist for integers a and b , because the expresion will get x 2 + 5 x + c for c an odd number, so the discriminant won't be a perfect square, so − 4 is the smallest integer which makes x 2 + 5 x + 6 be a prime number.
NO THIS DOESNT SEEM TO BE A PROPER APPROACH TO THE PROBLEM BECAUSE WE COULD HAVE HAD YET SMALLER VALUE OF X FOR A LARGER VALUE OF PRIME NUMBER( WHICH HERE WE OBVIOUSLY DONT HAVE)!!! PLEASE EXPLAIN THE LOGIN BEHIND,
WHY YOU STARTED WITH SMALLEST PRIME NO AND EVENTUALLY WHEN YOU GOT -4 AS ANSWER YOU ACCEPTED IT WITHOUT ANY DOUBT OF EXISTENCE OF A SMALLER VALUE OF X(SMALLER THAN -4)???
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Sorry about that, here I wrote the explanation.
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thanks for solution. I understood how you concluded -4 to be the an..but could you please patiently reply for why the discriminant wont be a perfect square if c is odd????
Please don't use "capitalization" as it is considered rude and aggressive...Noted!
The question should say "least integer", not "smallest integer" which is an ambiguous term.
Hooray, my answer was technically correct!
Sir, when we can differentiate between +4 and -4,and we know that -4 is less, why we need to bother about least or smallest? please explain the concept behind LEAST and SMALLEST INTEGER with proper example!!
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I'll assume your question is not intentionally ironic (though it appears to be deliberately histrionic). You just said that -4 is "less" than 4: the word "least" is literally the superlative of the comparative "less". This indeed makes "least integer" the most precise term to use in this question (but English has many edge cases).
Ask yourself, would you say that 0 . 0 0 0 0 1 is a small number? What about − 1 0 0 0 0 0 ? Which of those two is the smallest? I would say it depends on what you mean by "small". Yet the lesser one is clearly the negative one. Using "smallest" creates an inappropriate ambiguity in the question, but "least" is mathematically unambiguous.
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Sir, I am sorry if you perceived it as a deliberate attempt to taunt, but then if that was the case i would not have asked you to explain it with suitable example.Given that you are 41 and ofcourse experienced enough to satisfy my curious mind I asked it that way!!! Thanks for explanation sir. So by SMALL you mean the magnitude of number alone, but by LEAST the value of the number with its sign is considered.
The expression is always even, and the only even prime is 2, so we just have to solve
x 2 + 5 x + 6 = 2
( x + 4 ) ( x + 1 ) = 0
x = − 4 ∨ x = − 1
The smallest is x = − 4
We first notice that the equation always equals an even number for integer values of x. If x is odd, then you have ODD + ODD + EVEN = EVEN. If x is even, then you have EVEN + EVEN + EVEN=EVEN. We know that the only prime number that is even is 2, so we check if x 2 + 5 x + 6 = 2 has integer solutions, otherwise there are no solutions. We can factor this into ( x + 1 ) ( x + 4 ) = 0 , which gives us solutions of x = − 1 or x = − 4 . Our answer is − 4
I think the equation could be written as x(x+5)=n-6. If n is a prime number but not 2, it cannot be even. However, x(x+5) is always an even number, thus x(x+5)+6 must be even. So n must be 2. When n=2, x can be -4 or -1. -4 is the answer since it is smaller than -1.
The question is to find the smallest integer value. So the corresponding prime should be the smallest one that is 2. This implies:
X2 + 5x + 6 = 2
X2 + 5x + 4 =0
X2 + 4x +x + 4 =0
This gives x= -1 and x=-4, but the smallest integer is -4, implies the answer is -4.
I solve it using my guessing. Let f ( x ) = x 2 + 5 x + 6
The smallest prime number is 2, therefore let f ( x ) = 2 , we have the equation: x 2 + 5 x + 6 = 2
Solving that equation, we came out with 2 solutions: x = − 4 and x = − 1 .
Hence, the answer would be − 4 .
I don't know what to do if f ( x ) = 2 and the equation have no solution.
Let's Factor x^2 + 5x + 6 as:
(1) (x+2)(x+3)
(2) x (x+5) + 6.
Notice that in (2), x(x+5) is always even whatever the value of x is. The whole expression (2) is always even since it is the sum of two even numbers.
Since 2 is the only even prime number, x^2 + 5x + 6 = 2. Solving this quadratic equation, x= -1, -4.
Therefore, the smallest possible value of x is -4.
Quadratic formula is not so useless huh. Haha joke lang.
Notice that x²+5x+6 is always an even number, so the only way to be prime is to be 2. Solving the equation x²+5x+6=2 we get x=-4, x=-1. Since -4 is smaller this is our solution.
equivalent to the problem: find a prime p such that 1+4p is a 2, and a is an integer. There is only one prime (2) that satisfies this requirement, thus 1+4 2 = 9. so find the solutions of the equation: x 2 +5 x + 6 = 2 x=(-1, -4) hence -4 is the solution.
x 2 + 5 x + 6 = x 2 + 2 x + 3 x + 6 . = x ( x + 2 ) + 3 ( x + 2 ) = ( x + 2 ) ( x + 3 ) since the prime factorization of a prime number is 1 and the prime number itself, so either ( x + 3 ) or ( x + 2 ) is equal to 1. if any of these expressions is equal to one, then the other one would be negative. and a positive times a negative is a negative, which can never be a prime number. so , either ( x + 3 ) or ( x + 2 ) is -1. to yield the lowest possible integer, ( x + 3 ) = − 1 which gives x = − 4
-1 is also an answer. But, -4 < -1
But the question indicates 'smaller' so I think magnitude can be considered.
x 2 + 5 x + 6 = ( x + 2 ) ( x + 3 ) Assuming x is an integer, x + 2 and x + 3 must both be integers, and their product must be prime as per the problem: this implies ∣ x + 2 ∣ = 1 or ∣ x + 3 ∣ = 1 , or otherwise the product is composite. So the smallest possibly satisfactory integer value of x satisfies x + 3 = − 1 x = − 4
Indeed, if x = − 4 , then the result is 2 , which is prime.