Questions: How many solutions are there to the system of equations (y=-3 x^2+2) and (y=4 x^2+3) ? (Enter DNE if the answer doesn't exist.)

How many solutions are there to the system of equations (y=-3 x^2+2) and (y=4 x^2+3) ? (Enter DNE if the answer doesn't exist.)
Transcript text: How many solutions are there to the system of equations $y=-3 x^{2}+2$ and $y=4 x^{2}+3$ ? (Enter DNE if the answer doesn't exist.)
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Solution

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Solution Steps

Step 1: Formulate the problem

Given two quadratic equations $y = ax^2 + b$ and $y = cx^2 + d$, we set them equal to each other to get the equation: $(-3 - 4)x^2 + (2 - 3) = 0$.

Step 2: Solve the equation using the discriminant

The discriminant of the equation is calculated as $\Delta = 1$. Since $\Delta > 0$, there are two distinct real solutions. This means the curves intersect at two points.

Final Answer: The number of solutions is 2.

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