please help with this function

Question

if f(x)= 4x^3+Ax^2+7x-3 and f(2)=1, what is the value of A?

Answer 1

A general rule of solving these types of problems is that the number of unknowns is equal to the number of equations needed to solve it. Thankfully, we have 1 unknown, A, and one equation, f(2) = 1

So, the first step is to actually plug 2 in for x in the big equation. Once you do this, you will get:

f(2) = 4(2)³ + A(2)² + 7(2) - 3

Well, using our knowledge that:

1) f(2) = 1,

2) f(2) = 4(2)³ + A(2)² + 7(2) - 3

We can write that

1 = 4(2)³ + A(2)² + 7(2) - 3

Because both are equal to f(2)

Now, multiplying everything out, we get: 4A +32 +14 - 3 = 1

Simplifying it, we now have: 4A = -42

Now dividing, we get that A = -10.5

I hope this is understandable. If you need more explanation, just let me know.

Answer 2

For f(x) = 4x³ + Ax² + 7x − 3, write f(2) = 4(2)³ + A(2)² + 7(2) − 3, equal to 32 + 4A + 14 − 3 or 43 + 4A, which is given as equal to 1.

43 + 4A = 1 forces 4A to -42, yielding A = -10.5.