(4.35E-2 - x)(7.56E-2 -x )
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(0.324 + 2x)2
Rearrange to get an expression of the form ax2 + bx + c = 0 and use the quadratic formula (solve for x. This gives:
x = 1.05E-2, 0.143
basically i understand they use the quadratic equation but how did they change that crazy fraction into the quadratic formula?
If that fraction = 0 (you left that out) then the denominator is not part of the problem unless it is zero.
32.88 E-4 - 11.91 E-2 x + x^2 = 0
x = [ 11.91E-2 +/- sqrt(142 E-4 - 132 E-4) ] / 2
= [ .119 +/- E-2 sqrt(10) ]/2
= [.119 +/- .031 ]/2
= .075 or -.044
for the following population of n=9 scores below.
11 12 16 15 19 16 13 17 11
range........
variance standard deviation s:
To understand how the given expression is rearranged into the quadratic formula, let's break it down step by step.
The expression we have is:
(4.35E-2 - x)(7.56E-2 - x) / (0.324 + 2x)^2
To simplify this expression, first, let's expand the numerator:
(0.0435 - x)(0.0756 - x)
Next, let's expand the denominator:
(0.324 + 2x)(0.324 + 2x)
Now we can rewrite the original expression as:
[(0.0435 - x)(0.0756 - x)] / [(0.324 + 2x)(0.324 + 2x)]
To proceed further, we can distribute and simplify:
(0.0435 × 0.0756 - x × 0.0756 - 0.0435 × x + x^2) / (0.324 × 0.324 + 0.324 × 2x + 0.324 × 2x + 2x × 2x)
Now, let's simplify each term:
(0.0032966 - 0.0756x - 0.0435x + x^2) / (0.104976 + 0.648x + 0.648x + 4x^2)
Combining like terms:
(0.0032966 - 0.1191x + x^2) / (0.104976 + 1.296x + 4x^2)
Now, we can rewrite this expression in the form of ax^2 + bx + c = 0, by multiplying both the numerator and denominator by the denominator (to clear the fraction):
(0.0032966 - 0.1191x + x^2) × (0.104976 + 1.296x + 4x^2) = 0
Multiplying and simplifying further:
0.0003460411 + 0.0042601376x + 0.013440576x^2 - 0.0049081356x^3 - 0.0604805736x^2 + 0.195830072x^4 + 0.0042601376^2 + 0.0520801456*x^3 + 0.169640208x^4 + 0.520726296x^5 = 0
Combining like terms and rearranging in descending order of exponents:
0.195830072x^4 + 0.169640208x^4 + 0.520726296x^5 - 0.0049081356x^3 + 0.0520801456*x^3 - 0.0604805736x^2 + 0.013440576x^2 + 0.0042601376x + 0.0003460411 + 0.0042601376^2 = 0
Finally, we have the equation:
0.68947028x^5 + 0.0471719112x^4 - 0.0118274328x^3 - 0.0471194366x^2 + 0.0042601376x + 0.00485016251097 = 0
Now, we can solve this quadratic equation using the quadratic formula to find the values of x.