f(x) = x^2 + 4x + k = 12; f(-6) = 0; if k is a constant and n is the

f(x)=x2+4x+k=12 , f(−6)=0 this results in k=0

so for f(x)=0, then x has to be '0'

OA:C

We are given that f(-6) = 0 which means that -6 is a solution of the function f(x).

f(-6) = (-6)^2 + 4*(-6) + k - 12 = 0.
from here, we get k = 0.

Now we need to find the value of n such that f(n) = 0, which means we need to find the second solution of f(x)

now, f(x) = x^2 + 4x - 12 = 0
so f(x) = (x+6)(x-2).

The other solution for f(x) = 2 = n such that f(n) or f(2) = 0

IMO, B

f(x) = x^2 + 4x + k = 12
f(-6) = x^2 + 4x + k = 0, 36-24+k=0, k=-12
f(n) = x^2 + 4x + k = 0, x^2 + 4x + (-12) = 0,
(x+6)(x-2)=0, n=2

Ans (B)

f(x)=x^2+4x+k=12; f(-6)=0. We are to determine n such that f(n)=0
From the given information, f(-6)=0
f(x=-6)=36-24+k=0, k=-12
f(x)=x^2+4x-12.
To find n, we need to equate f(x)=0 and find the roots. the second root has to be n. We already know one root of f(x), x=-6.
x^2+4x-12=0
(x+6)(x-2)=0
So the roots of f(x) are -6, and 2.
n is therefore 2.

The answer is B.

\(f(x)=x^{2}+4x+k=12\);
f(−6)=0;
--> if k is a constant and n is the number for which f(n)=0, what is the value of n?

\(f(−6)=x^{2}+4x+k -12=36 -24+ k- 12=0\)
--> k=0.

\(f(x)=x^{2}+4x-12=0\);
--> \(f(n)= n^{2} +4n -12=0\)
(n+6)*(n-2)=0
--> n=-6 and n=2

The answer is B.

OFFICIAL EXPLANATION

There are two main approaches you can take to solving this problem. You can either solve for k and turn this problem into a simple factoring problem or you can ignore k and reverse factor.

Method 1: Solve for K.

\(f(x) = x^2 + 4x + k = 12\)
\(f(x) = x^2 + 4x + k - 12 = 0\) (subtract 12)

Plug in \(x=-6\) knowing that the value of f(x) must equal zero since \(f(-6)=0\).
\(f(-6) = (-6)^2 + 4(-6) + k - 12 = 0\)
\(36 -24 + k - 12 = 0\)
\(0 - k = 0\) --> \(k = 0\)

The equation is now:
\(f(x) = x^2 + 4x + 0 - 12 = 0\)
\(f(x) = x^2 + 4x - 12 = 0\)

By factoring: \(x^2 + 4x - 12 = 0\) equals:
\((x+6)(x+n)\) Note: \(x+6\) is from \(f(-6)=0\)

Since we need two numbers that add to +4 and multiply to -12, we know that +6 and -2 work. Consequently, (x-2) is a factor and therefore, \(f(+2)=0\), so \(n = 2\).


Method 2: Ignore K.
A crucial insight in unlocking this problem is recognizing that if f(a) = 0 and f(x) is a quadratic equation in the form \(x^2 + bx + c = 0\), \((x – a)\) is a factor of the equation. In order to get the equation into quadratic form, subtract 12.
\(f(x) = x^2 + 4x + k = 12\)
\(f(x) = x^2 + 4x + k – 12 = 0\)

Since \(f(-6) = 0\), \((x+6)\) is a factor.

It is important to remember how factoring works. Specifically, remember the following:
\((x + d)(x + e) = x^2 + dex + de = 0\)
So: \((x + 6)(x + a) = x^2 + 4x + (k – 12)\)

With this in mind, you know that \(6 + a = 4\). So, \(a = -2\) and therefore, \((x - 2)\) is the other factor of the quadratic. So, \(f(2) = 0\) and \(n = 2\).

Answer B is correct.

There are two serious issues with the question. First, if a = b = c, then a = c is always true. But here, we are told f(x) = x^2 + 4x +k = 12, which means f(x) = 12. But then f(x) is never equal to zero (it is always equal to 12) so the situation described later is impossible. As written, the question isn't defining a function properly (a function can't be equal to two different things), and it only makes sense if the second equals sign in "f(x) = x^2 + 4x + k = 12" is actually a minus sign.

Then there's a language issue. The question asks for "the value of n" for which f(n) = 0, which means there's only one such value. But ignoring everything else in the question, it tells us f(-6) = 0, so "the value of n" is clearly -6, if there is only one value of n. The question would need to ask what n could be equal to, or needs to ask for the value of n that is not equal to -6.