Answer: Choice C
f(x) = 5x^2 + 25x + 30
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Explanation:
The roots, aka x intercepts, of this curve are x = -3 and x = -2. This is where the graph crosses the x axis.
Since x = -3 is a root, this makes x+3 a factor of the quadratic. Similarly, x = -2 leads to x+2 as another factor. I'm using the zero product property.
So far we have found that the polynomial is (x+3)(x+2). This isn't the full factorization because if we plugged x = -1 into that expression, then we would get
y = (x+3)(x+2)
y = (-1+3)(-1+2)
y = (2)(1)
y = 2
But we want y = 10 instead. So we must multiply that factorization by 5 to jump from 2 to 10 (i.e. 5*2 = 10)
Therefore, the full factorization of this parabola is y = 5(x+3)(x+2)
Now let's expand everything out and simplify
y = 5(x+3)(x+2)
y = 5(x^2+2x+3x+6)
y = 5(x^2+5x+6)
y = 5x^2+5*5x+5*6
y = 5x^2 + 25x + 30
Choice C is the final answer
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To check this, we can plug in x = -3 and we should get 0
y = 5x^2 + 25x + 30
y = 5(-3)^2 + 25(-3) + 30
y = 5(9) + 25(-3) + 30
y = 45 - 75 + 30
y = -30 + 30
y = 0
This proves that x = -3 is a root of y = 5x^2 + 25x + 30
I'll let you check x = -2. You should also get y = 0 when plugging this x value in.
Plugging x = -1 should lead to y = 10 as the last bit of confirmation. I'll let you check this one as well.
