Let f(x) = Ax² + Bx + C where A, B, C...

MathematicsMaximum and Minimum Values of Quadratic ExpressionsJEE Advanced 1998Moderate

Let $f (x) = A x^{2} + B x + C$ where $A, B, C$ are real numbers. Prove that if $f (x)$ is an integer whenever $x$ is an integer, then the numbers $2 A, A + B$ and $C$ are all integers. Conversely, prove that if the numbers $2 A, A + B$ and $C$ are all integers then $f (x)$ is an integer whenever $x$ is an integer.

Visualized Solution (English)

Understanding the Quadratic Function .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f (x)

Given function: $f (x) = A x^{2} + B x + C$
Condition: $f (x) \in Z$ for all $x \in Z$
Goal (Part 1): Prove $2 A, A + B, C \in Z$

Evaluating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f (0) to find .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C

Substitute $x = 0$ into $f (x)$ :
$f (0) = A (0)^{2} + B (0) + C = C$
Since $f (0)$ is an integer, $C$ must be an integer.

Evaluating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f (1) and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f (- 1)

Substitute $x = 1$ :
$f (1) = A (1)^{2} + B (1) + C = A + B + C$
Substitute $x = - 1$ :
$f (- 1) = A (- 1)^{2} + B (- 1) + C = A - B + C$
Both $f (1)$ and $f (- 1)$ are integers.

Proving .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A + B and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 2 A are Integers

To find $A + B$ :
$A + B = f (1) - C$
Since $f (1) \in Z$ and $C \in Z$ , $A + B$ is an integer.
To find $2 A$ :
$f (1) + f (- 1) = (A + B + C) + (A - B + C) = 2 A + 2 C$
$2 A = (f (1) + f (- 1)) - 2 C$
Since $f (1), f (- 1), C \in Z$ , $2 A$ is an integer.

The Converse: Rewriting .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f (x)

Given: $2 A, A + B, C \in Z$
Goal (Part 2): Prove $f (x) \in Z$ for all $x \in Z$
Rearrange $f (x)$ :
$f (x) = A x^{2} - A x + A x + B x + C$
$f (x) = A (x^{2} - x) + (A + B) x + C$
$f (x) = 2 A (\frac{x ( x - 1 )}{2}) + (A + B) x + C$

Final Proof using Integer Properties

Analyze the terms for $x \in Z$ :
1. $x (x - 1)$ is a product of two consecutive integers, so it is always even.
2. Therefore, $\frac{x ( x - 1 )}{2}$ is always an integer.
3. Since $2 A, A + B, C$ are integers, each term in the sum is an integer:
$\in Z 2 A \cdot \in Z \frac{x ( x - 1 )}{2} + \in Z (A + B) \cdot \in Z x + \in Z C$
Conclusion: $f (x)$ is the sum of three integers, so $f (x)$ is an integer.

00:00 / 00:00

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Visualized Solution (English)

Understanding the Quadratic Function $f (x)$

Evaluating $f (0)$ to find $C$

Evaluating $f (1)$ and $f (- 1)$

Proving $A + B$ and $2 A$ are Integers

The Converse: Rewriting $f (x)$

Final Proof using Integer Properties

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Visualized Solution (English)

Understanding the Quadratic Function .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f(x)

Evaluating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f(0) to find .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C

Evaluating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f(1) and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f(−1)

Proving .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A+B and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 2A are Integers

The Converse: Rewriting .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f(x)

Final Proof using Integer Properties

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Understanding the Quadratic Function $f (x)$

Evaluating $f (0)$ to find $C$

Evaluating $f (1)$ and $f (- 1)$

Proving $A + B$ and $2 A$ are Integers

The Converse: Rewriting $f (x)$

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