What is 4 is to 3 as x is to 20
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Hey there! In this problem, the solutions, both real and imaginary are provided. We have to work backwards to create a polynomial function which matches these solutions:
f(x) = (x - 6), (x + [5 + 2i]), (x + [5 - 2i])
Create a difference of squares,
f(x) = (x - 6), ([x + 5] + 2i), ([x + 5] - 2i)
f(x) = (x - 6), ([x + 5]^2 + 4)
[(x + 5)(x + 5)] + 4
x^2 + 5x + 5x + 25 + 4
x^2 + 10x + 29
f(x) = (x-6), (x^2 + 10x + 29)
f(x) = x^3 + 10x^2 + 29x - 6x^2 - 60x - 174
Combine like terms,
f(x) = x^3 + 4x^2 + 29x - 174
There we have our answer, a polynomial of degree 3.
1.You have two ways for solving this.
First, you could use a graphing utility or calculator to graph both functions on the same plane, and given that the functions do not have a
coordinate, the solution of 
will be the 
coordinate in which the tow graphs intercept (cross each other) as you can see in the picture (1).
The other way is equating the two equations and solve for x:
We know that
and 
, so is equivalent to 
. The only thing left is solving the equation for x:
As you can see, both ways gives us the same answer: the solution to the equation is .
2. For this one we can repeat the exact same procedures as before:
Graph the functions on the same plane using a graph calculator or graphing utility, and identify the coordinate in which the two graphs intercept as you can see in the picture (2), or you we can also equate both equations, and solve for :
And just like before, both ways gives us the same answer: the solution to the equation is
So, the answer that you should enter in your boxes are: 2 in the first, and 1 in the second one.
First, you could use a graphing utility or calculator to graph both functions on the same plane, and given that the functions do not have a
The other way is equating the two equations and solve for x:
We know that
As you can see, both ways gives us the same answer: the solution to the equation
2. For this one we can repeat the exact same procedures as before:
Graph the functions on the same plane using a graph calculator or graphing utility, and identify the
And just like before, both ways gives us the same answer: the solution to the equation
So, the answer that you should enter in your boxes are: 2 in the first, and 1 in the second one.