If f left parenthesis x right parenthesis equals 2 x cubed plus Ax squared plus 7 x minus 7 and f left parenthesis 2 right parenthesis equals 7​, what is the value of​ A?

We are given the function:

\[
f(x) = 2x^3 + Ax^2 + 7x - 7
\]

And we know that:

\[
f(2) = 7
\]

Substitute \(x = 2\) into the function:

\[
f(2) = 2(2)^3 + A(2)^2 + 7(2) - 7
\]

Simplify the expression:

\[
f(2) = 2(8) + A(4) + 14 - 7
\]

\[
f(2) = 16 + 4A + 14 - 7
\]

\[
f(2) = 23 + 4A
\]

We are told that \(f(2) = 7\), so:

\[
23 + 4A = 7
\]

Solve for \(A\):

\[
4A = 7 - 23
\]

\[
4A = -16
\]

\[
A = \frac{-16}{4} = -4
\]

Thus, the value of \(A\) is \(-4\).

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you solve cubic equations in general?
2. What are other ways to find coefficients in a polynomial?
3. How would you approach solving a system of equations involving multiple variables?
4. What happens when a function’s value is undefined for a given \(x\)?
5. How do the coefficients of a polynomial affect its graph?

**Tip:** When solving equations, always simplify expressions step by step to avoid errors.

If f(x) equals 2x cubed plus Ax squared plus 7x minus 7 and f(2) equals 7, what is the value of A?

Learn how to find the value of A in the polynomial function f(x) = 2x^3 + Ax^2 + 7x - 7 when f(2) equals 7. This problem involves polynomial equations, substitution, and basic algebraic concepts, suitable for students in Grades 9-11.

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