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Two integers have a sum of -1 and a product of -56 . What is the positive difference between the two integers?

Two integers have a sum of $-1$ and a product of $-56$ . What is the positive difference between the two integers?

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Write a linear equation from two points

Full solution

Q. Two integers have a sum of

-1

and a product of

-56

. What is the positive difference between the two integers?

Define Integers $x$ and $y$ : Let's call the two integers $x$ and $y$ . We know that $x + y = -1$ and $xy = -56$ .
Create Quadratic Equation: We can use the sum and product to create a quadratic equation. If $x$ and $y$ are roots of the equation, then it can be written as $t^2 - (x + y)t + xy = 0$ . Substituting the given values, we get $t^2 + t - 56 = 0$ .
Factor the Quadratic Equation: Now, we need to factor the quadratic equation. Factors of $-56$ that add up to $1$ are $7$ and $-8$ . So, the equation factors to $(t - 7)(t + 8) = 0$ .
Set Factors Equal to Zero: Setting each factor equal to zero gives us the integers. $t - 7 = 0$ , so $t = 7$ . $t + 8 = 0$ , so $t = -8$ .
Calculate Positive Difference: The positive difference between $7$ and $-8$ is $|7 - (-8)| = |7 + 8| = 15$ .

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