A spirit shop outside the hockey stadium sells merchandise representing the three school districts that built the arena. The ent
rances to the arena, the spirit shop, and the parking lot form the vertices Of a triangle. The planning committee has decided to erect a memorial statue and would like it to be equidistant from the entrances to the arena, spirit shop, and the parking lot. At what point should the statue be located? Explain your answer.
1 answer:
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The formula for calculating length is:
We can also write
or 
Why it does not matter?
Let's assume we have 2 numbers, a and b.
When we perform a subtraction:
, we get another number 
When we perform another subtraction:
, we get a number 
When we raise or to the power of 2, the result is the same, 
.
We can also write
Why it does not matter?
Let's assume we have 2 numbers, a and b.
When we perform a subtraction:
When we perform another subtraction:
When we raise
Answer:
{1, (-1±√17)/2}
Step-by-step explanation:
There are formulas for the real and/or complex roots of a cubic, but they are so complicated that they are rarely used. Instead, various other strategies are employed. My favorite is the simplest--let a graphing calculator show you the zeros.
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Descartes observed that the sign changes in the coefficients can tell you the number of real roots. This expression has two sign changes (+-+), so has 0 or 2 positive real roots. If the odd-degree terms have their signs changed, there is only one sign change (-++), so one negative real root.
It can also be informative to add the coefficients in both cases--as is, and with the odd-degree term signs changed. Here, the sum is zero in the first case, so we know immediately that x=1 is a zero of the expression. That is sufficient to help us reduce the problem to finding the zeros of the remaining quadratic factor.
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Using synthetic division (or polynomial long division) to factor out x-1 (after removing the common factor of 4), we find the remaining quadratic factor to be x²+x-4.
The zeros of this quadratic factor can be found using the quadratic formula:
a=1, b=1, c=-4
x = (-b±√(b²-4ac))/(2a) = (-1±√1+16)/2
x = (-1 ±√17)2
The zeros are 1 and (-1±√17)/2.
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The graph shows the zeros of the expression. It also shows the quadratic after dividing out the factor (x-1). The vertex of that quadratic can be used to find the remaining solutions exactly: -0.5 ± √4.25.
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The given expression factors as ...
4(x -1)(x² +x -4)
