a.
The polynomial w^2+18w+84 cannot be factored
The perfect square trinomial is w^2+18w + 81
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The reason the original can't be factored is that solving w^2+18w+84=0 leads to no real solutions. Use the quadratic formula to see this. The graph of y = x^2+18x+84 shows there are no x intercepts. A solution and an x intercept are basically the same. The x intercept visually represents the solution.
w^2+18w+81 factors to (w+9)^2 which is the same as (w+9)(w+9). We can note that w^2+18w+81 is in the form a^2+2ab+b^2 with a = w and b = 9
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b.
The polynomial y^2-10y+23 cannot be factored
The perfect square trinomial is y^2-10y + 25
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Using the quadratic formula, y^2-10y+23 = 0 has no rational solutions. The two irrational solutions mean that we can't factor over the rationals. Put another way, there are no two whole numbers such that they multiply to 23 and add to -10 at the same time.
If we want to complete the square for y^2-10y, we take half of the -10 to get -5, then square this to get 25. Therefore, y^2-10y+25 is a perfect square and it factors to (y-5)^2 or (y-5)(y-5)
Easy, since you have the diameter, which is very needed in order to find the circumference, you can use the equation d x 3.14 = c. d is the diameter and c is the circumference to 300 x 3.14 = 942 ft
Simultaneous Equations
3x + y = 4
5x - y = 22
The signs of the matching values aren't the same so we add the rest of the equations.
8x = 26
Divide both sides by 8 to get x = 3.25
Now substitute back into the first equation.
(3 x 3.25) + y = 4
9.75 + y = 4
Subtract 9.75 from both sides
y = -5.75
