The total area of the two boards is represented by the expression 7x^2 - 6x + 2 square inches.
Let A be the area of the second board in square inches.
Therefore, the expression representing the area of the first board is 7x^2 - 6x + 2 - A.
Since the area of the first board is 2x^2 - 9x + 8 square inches, we have the equation:
2x^2 - 9x + 8 = 7x^2 - 6x + 2 - A.
Simplifying the equation, we get:
5x^2 - 3x + 6 - A = 0.
To find the maximum area of the second board, we need to find the maximum value of A.
Since the coefficient of the x^2 term is positive, the parabola opens upward, which means that its vertex represents the minimum value of the function.
The x-coordinate of the vertex of the quadratic equation, represented by f(x) = 5x^2 - 3x + 6 - A, is given by x = -b/(2a), where a is the coefficient of the x^2 term, and b is the coefficient of the x term.
In this case, a = 5 and b = -3, so the x-coordinate of the vertex is x = -(-3)/(2(5)) = 3/10.
Substituting this x-coordinate into the original equation, we get:
f(3/10) = 5(3/10)^2 - 3(3/10) + 6 - A.
Simplifying the equation f(3/10) = -3/25 - 9/10 + 6 - A, we have:
-3/25 - 9/10 + 6 - A = 0.
Combining like terms, we get:
-33/50 - A = 0.
Adding A to both sides of the equation, we have:
-33/50 + A = 0 + A.
Therefore, the maximum value of A is 33/50, which is approximately equal to 0.66.
Thus, the maximum area of the second board is approximately 0.66 square inches.
Hunter and Brian are creating a poster for a project and they need to combine two different size boards to create their project. If the total area cannot exceed 7x^2 - 6x + 2 square inches and one board is 2x^2 - 9x + 8 square inches, then what must be the maximum area of the second board?
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