The formula N = 2Xsquared + 22X + 320 represents the number of inmates, N, in thousands, in U.S. state and federal prisons x years after 1980. According to the formula, in what year were there 740 thousand inmates in the U.S. state and federal prisons?
The formula N = 2Xsquared + 22X + 320 represents the number of inmates, N, in thousands, in U.S. state and federal prisons x years after 1980. According to the formula, in what year were there 740 thousand inmates in the U.S. state and federal prisons?
2x^2 + 22x + 320 = 740
2x^2 + 22x - 420 = 0
x^2 + 11x - 210 = 0
(x - 10)(x + 21)
10 years
To determine the year when there were 740,000 inmates in U.S. state and federal prisons according to the given formula N = 2X^2 + 22X + 320, we need to solve the equation for X.
The equation can be rewritten as:
2X^2 + 22X + 320 = 740.
First, let's subtract 740 from both sides of the equation to isolate the terms on one side:
2X^2 + 22X + 320 - 740 = 0.
Now, simplify the equation:
2X^2 + 22X - 420 = 0.
Next, we can use the quadratic formula to solve for X:
X = (-b ± √(b^2 - 4ac)) / (2a),
where a = 2, b = 22, and c = -420.
Substituting these values into the quadratic formula, we get:
X = (-22 ± √(22^2 - 4(2)(-420))) / (2(2)).
Simplifying further:
X = (-22 ± √(484 + 3360)) / 4,
X = (-22 ± √3844) / 4.
Now, computing the square root of 3844:
X = (-22 ± 62) / 4.
We will calculate both roots:
X1 = (-22 + 62) / 4 = 40 / 4 = 10,
X2 = (-22 - 62) / 4 = -84 / 4 = -21.
Since we are looking for the number of years after 1980, X1 = 10 represents the year 1990.
Therefore, in the year 1990, there were 740,000 inmates in U.S. state and federal prisons.