Question

PLS ANSWER RN

For what real values of $c$ is $x^2 - 8x - 4x + c$ the square of a binomial?

Answer 1

Answer:

Step-by-step explanation:

To determine the values of c for which the expression is a square of a binomial, we'll apply the square of a binomial pattern:

(a + b)^2 = a^2 + 2ab + b^2

(a - b)^2 = a^2 - 2ab + b^2

Looking at the given expression, x^2 - 8x - 4x + c, we can make a few observations:

1. The first term is x^2, which suggests that the first term in the binomial is x.

2. The middle term is -12x, which indicates that the two terms in the binomial are being subtracted.

Now, we can rewrite the expression as:

x^2 - 12x + c

For this to be a square of a binomial, the middle term (-12x) should be twice the product of the first and second terms of the binomial, and the last term (c) should be the square of the second term.

Applying the pattern (a - b)^2:

a^2 - 2ab + b^2 = x^2 - 12x + c

We can see that:

2ab = 12x

b^2 = c

Solving for b:

b = 6

Finding c:

c = b^2 = 6^2 = 36

Therefore, the only real value of c for which x^2 - 8x - 4x + c is the square of a binomial is c = 36.

Answer 2

Answer:

$\sf c = \boxed{\,\, 36 \, \, }$

Step-by-step explanation:

To find the real values of $\sf c$ for which $\sf x^2 - 8x - 4x + c$ is the square of a binomial, let's complete the square.

The given expression is:

$\sf x^2 - 8x - 4x + c$

Combine like terms:

$\sf x^2 - 12x + c$

Now, to make it a perfect square trinomial, add and subtract $\sf (\textsf{half of the coefficient of } x)^2$, which is $\sf (-6)^2 = 36$:

$\sf x^2 - 12x + 36 + c - 36$

Factor the perfect square trinomial:

$\sf (x - 6)^2 + (c - 36)$

Now, for this expression to be the square of a binomial, $\sf c - 36$ must be equal to 0, because we want to remove the $\sf c$ term:

$\sf c - 36 = 0$

Solve for $\sf c$:

$\sf c = 36$

So, the real value of $\sf c$ for which $\sf x^2 - 8x - 4x + c$ is the square of a binomial is:

$\sf c = \boxed{\,\, 36 \, \, }$