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In the xy-plane, the graph of y = 3x^2 −...

In the xy-plane, the graph of `y = 3x^2 − 14x` intersects the graph of y = x at the points (0, 0) and (a, a). What is the value of a ?

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The intersection points of the graphs of `y = 3x^2 − 14x and y = x` can be found by solving the system consisting of these two equations. To solve the system, substitute x for y in the first equation. This gives` x = 3x^2 − 14x`. Subtracting x from both sides of the equation gives `0 = 3x^2 − 15x`. Factoring 3x out of each term on the left-hand side of the equation gives 0 = 3x(x − 5). Therefore, the possible values for x are 0 and 5. Since y = x, the two intersection points are (0, 0) and (5, 5). Therefore, a = 5.
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