It seems like you're trying to find a quadratic equation given some specific values. It looks like you have two points that might define this equation, possibly indicating an input-output relationship (like a tax calculation).
From your message, it seems like you have two pairs of values for \( y \):
- When \( x = -1 \), \( y = 10 \)
- When \( x = 2 \), \( y = 2160 \)
You are looking for a quadratic equation in the form of \( y = ax^2 + bx + c \). To find the coefficients \( a \), \( b \), and \( c \), we can set up a system of equations based on these points.
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For \( x = -1 \): \[ 10 = a(-1)^2 + b(-1) + c \implies 10 = a - b + c \quad \text{(Equation 1)} \]
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For \( x = 2 \): \[ 2160 = a(2)^2 + b(2) + c \implies 2160 = 4a + 2b + c \quad \text{(Equation 2)} \]
To fully determine the equation, we usually need a third point. However, we can also make an assumption about the form of the quadratic equation to simplify it. Let’s assume \( y = ax^2 \), which would fit the general quadratic form, ignoring the linear \( bx + c \) component initially for simplicity.
From the point \( (2, 2160) \): \[ 2160 = a(2^2) \implies 2160 = 4a \implies a = 540 \]
The equation \( y = 540x^2 \) fits the first point to some degree since there is no linear term to influence it. To check it, we can substitute:
For \( x = -1 \): \[ y = 540(-1)^2 = 540 \] This does not match \( y = 10 \), indicating we need the linear terms.
Since you mentioned different formats in your prompt (“blank times blank squared”), we realize that we likely need a complete expression.
General Path Forward
- Form three simultaneous equations from the points, aiming for \( y = ax^2 + bx + c \).
- Check values and determine if you need to constrain your model further for reasoning (like whether \( c = 0 \) or fixing \( b \)).
The standard way to manage a quadratic here is:
- Calculate it exactly.
- Graph it and find coefficients or explore if further assumptions apply (via constraints).
If you want to follow up with another point or if you have more requirements (like \( c \) being zero), please clarify!