Let's represent the unknown negative number as "x."
According to the given information, the square of the number is equal to five more than one-half of the number. Mathematically, we can write this as:
x^2 = (1/2)x + 5
To find the value of x, we need to solve this quadratic equation. Rearranging the equation, we have:
x^2 - (1/2)x - 5 = 0
To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 1, b = -1/2, and c = -5. Substituting these values into the quadratic formula:
x = (-(1/2) ± √((-1/2)^2 - 4(1)(-5))) / (2(1))
Simplifying further:
x = (-1/2 ± √(1/4 + 20)) / 2
x = (-1/2 ± √(81/4)) / 2
x = (-1/2 ± 9/2) / 2
x = (-1 ± 9) / 4
Therefore, we have two possible solutions for x:
x = (-1 + 9) / 4 = 8/4 = 2
x = (-1 - 9) / 4 = -10/4 = -5/2
Since we are looking for a negative number, the solution is x = -5/2.
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