Question #6770

Examples of quadratic equations whose roots are integers?

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Answer

Below is a list of quadratic equations whose roots are integers:

Positive discriminant (two distincts real roots)

• -5x² - 5x = 0 ===> x1 = 0, x2 = -1
• -5x² + 5 = 0 ===> x1 = 1, x2 = -1
• -5x² + 5x = 0 ===> x1 = 1, x2 = 0
• -4x² - 4x= 0 ===> x1 = 0, x2 = -1
• -4x² + 4 = 0 ===> x1 = 1, x2 = -1
• -4x² + 4x = 0 ===> x1 = 1, x2 = 0
• -3x² - 3x = 0 ===> x1 = 0, x2 = -1
• -3x² + 3 = 0 ===> x1 = 1, x2 = -1
• -3x² + 3x = 0 ===> x1 = 1, x2 = 0
• -2x² - 4x = 0 ===> x1 = 0, x2 = -2
• -2x² - 2x = 0 ===> x1 = 0, x2 = -1
• -2x² - 2x+4 = 0 ===> x1 = 1, x2 = -2
• -2x² +2 = 0 ===> x1 = 1, x2 = -1
• -2x² + 2x = 0 ===> x1 = 1, x2 = 0
• -2x² +2x +4 = 0 ===> x1 = 2, x2 = -1
• -2x² + 4x = 0 ===> x1 = 2, x2 = 0
• -x² - 5x - 4 = 0 ===> x1 = -1, x2 = -4
• -x² - 5x = 0 ===> x1 = 0, x2 = -5
• -x² - 4x - 3 = 0 ===> x1 = -1, x2 = -3
• -x² - 4x = 0 ===> x1 = 0, x2 = -4
• -x² - 4x + 5 = 0 ===> x1 = 1, x2 = -5
• -x² - 3x - 2 = 0 ===> x1 = -1, x2 = -2
• -x² - 3x = 0 ===> x1 = 0, x2 = -3
• -x² - 3x + 4 = 0 ===> x1 = 1, x2 = -4
• -x² - 2x = 0 ===> x1 = 0, x2 = -2
• -x² - 2x + 3 = 0 ===> x1 = 1, x2 = -3
• -x² - 1x = 0 ===> x1 = 0, x2 = -1
• -x² - 1x + 2 = 0 ===> x1 = 1, x2 = -2
• -x² +1 = 0 ===> x1 = 1, x2 = -1
• -x² +4 = 0 ===> x1 = 2, x2 = -2
• -x² +1x = 0 ===> x1 = 1, x2 = 0
• -x² +1x +2 = 0 ===> x1 = 2, x2 = -1
• -x² + 2x = 0 ===> x1 = 2, x2 = 0
• -x² + 2x +3 = 0 ===> x1 = 3, x2 = -1
• -x² + 3x -2 = 0 ===> x1 = 2, x2 = 1
• -x² + 3x = 0 ===> x1 = 3, x2 = 0
• -x² + 3x +4 = 0 ===> x1 = 4, x2 = -1
• -x² + 4x -3 = 0 ===> x1 = 3, x2 = 1
• -x² + 4x = 0 ===> x1 = 4, x2 = 0
• -x² + 4x +5 = 0 ===> x1 = 5, x2 = -1
• -x² + 5x -4 = 0 ===> x1 = 4, x2 = 1
• -x² + 5x = 0 ===> x1 = 5, x2 = 0
• x² - 5x = 0 ===> x1 = 0, x2 = 5
• x² -5x+4 = 0 ===> x1 = 1, x2 = 4
• x² -4x-5 = 0 ===> x1 = -1, x2 = 5
• x² -4x+0 = 0 ===> x1 = 0, x2 = 4
• x² -4x+3 = 0 ===> x1 = 1, x2 = 3
• x² -3x-4 = 0 ===> x1 = -1, x2 = 4
• x² -3x+0 = 0 ===> x1 = 0, x2 = 3
• x² -3x+2 = 0 ===> x1 = 1, x2 = 2
• x² -2x-3 = 0 ===> x1 = -1, x2 = 3
• x² -2x+0 = 0 ===> x1 = 0, x2 = 2
• x² -1x-2 = 0 ===> x1 = -1, x2 = 2
• x² -1x = 0 ===> x1 = 0, x2 = 1
• x² - 4 = 0 ===> x1 = -2, x2 = 2
• x² - 1 = 0 ===> x1 = -1, x2 = 1
• x² + 1x - 2 = 0 ===> x1 = -2, x2 = 1
• x² + 1x = 0 ===> x1 = -1, x2 = 0
• x² + 2x - 3 = 0 ===> x1 = -3, x2 = 1
• x² + 2x = 0 ===> x1 = -2, x2 = 0
• x² + 3x - 4 = 0 ===> x1 = -4, x2 = 1
• x² + 3x = 0 ===> x1 = -3, x2 = 0
• x² + 3x + 2 = 0 ===> x1 = -2, x2 = -1
• x² + 4x - 5 = 0 ===> x1 = -5, x2 = 1
• x² + 4x = 0 ===> x1 = -4, x2 = 0
• x² + 4x + 3 = 0 ===> x1 = -3, x2 = -1
• x² + 5x = 0 ===> x1 = -5, x2 = 0
• x² + 5x + 4 = 0 ===> x1 = -4, x2 = -1
• 2x² - 4x = 0 ===> x1 = 0, x2 = 2
• 2x² - 2x - 4 = 0 ===> x1 = -1, x2 = 2
• 2x² -2x + 0 = 0 ===> x1 = 0, x2 = 1
• 2x² +0x - 2 = 0 ===> x1 = -1, x2 = 1
• 2x² +2x - 4 = 0 ===> x1 = -2, x2 = 1
• 2x² +2x = 0 ===> x1 = -1, x2 = 0
• 2x² +4x = 0 ===> x1 = -2, x2 = 0
• 3x² - 3x = 0 ===> x1 = 0, x2 = 1
• 3x² - 3 = 0 ===> x1 = -1, x2 = 1
• 3x² + 3x = 0 ===> x1 = -1, x2 = 0
• 4x² - 4x = 0 ===> x1 = 0, x2 = 1
• 4x² - 4 = 0 ===> x1 = -1, x2 = 1
• 4x² +4x = 0 ===> x1 = -1, x2 = 0
• 5x² - 5x = 0 ===> x1 = 0, x2 = 1
• 5x² - 5 = 0 ===> x1 = -1, x2 = 1
• 5x² +5x = 0 ===> x1 = -1, x2 = 0

Discriminant equal zero (two repeated real roots)

• -8x² = 0 ===> x1 = x2 = 0
• -7x² = 0 ===> x1 = x2 = 0
• -6x² = 0 ===> x1 = x2 = 0
• -5x² = 0 ===> x1 = x2 = 0
• -4x² - 8x - 4 = 0 ===> x1 = x2 = -1
• -4x² = 0 ===> x1 = x2 = 0
• -4x² + 8x - 4 = 0 ===> x1 = x2 = 1
• -3x² - 6x - 3 = 0 ===> x1 = x2 = -1
• -3x² = 0 ===> x1 = x2 = 0
• -3x² + 6x - 3 = 0 ===> x1 = x2 = 1
• -2x² - 8x - 8 = 0 ===> x1 = x2 = -2
• -2x² - 4x - 2 = 0 ===> x1 = x2 = -1
• -2x² = 0 ===> x1 = x2 = 0
• -2x² + 4x - 2 = 0 ===> x1 = x2 = 1
• -2x² + 8x - 8 = 0 ===> x1 = x2 = 2
• -x² - 4x - 4 = 0 ===> x1 = x2 = -2
• -x² - 2x - 1 = 0 ===> x1 = x2 = -1
• -x² = 0 ===> x1 = x2 = 0
• -x² + 2x - 1 = 0 ===> x1 = x2 = 1
• -x² + 4x - 4 = 0 ===> x1 = x2 = 2
• x² - 4x + 4 = 0 ===> x1 = x2 = 2
• x² - 2x + 1 = 0 ===> x1 = x2 = 1
• x² = 0 ===> x1 = x2 = 0
• x² + 2x + 1 = 0 ===> x1 = x2 = -1
• x² + 4x + 4 = 0 ===> x1 = x2 = -2
• 2x² - 8x + 8 = 0 ===> x1 = x2 = 2
• 2x² - 4x + 2 = 0 ===> x1 = x2 = 1
• 2x² = 0 ===> x1 = x2 = 0
• 2x² + 4x + 2 = 0 ===> x1 = x2 = -1
• 2x² + 8x + 8 = 0 ===> x1 = x2 = -2
• 3x² - 6x + 3 = 0 ===> x1 = x2 = 1
• 3x² = 0 ===> x1 = x2 = 0
• 3x² + 6x + 3 = 0 ===> x1 = x2 = -1
• 4x² - 8x + 4 = 0 ===> x1 = x2 = 1
• 4x² = 0 ===> x1 = x2 = 0
• 4x² + 8x + 4 = 0 ===> x1 = x2 = -1
• 5x² = 0 ===> x1 = x2 = 0
• 6x² = 0 ===> x1 = x2 = 0
• 7x² = 0 ===> x1 = x2 = 0
• 8x² = 0 ===> x1 = x2 = 0

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