I will give brainliest if your answer is correct :) Let a = x^2 + 4. Rewrite the following equation in terms of a and set it equal to zero. (x^2 + 4)^2 + 32 = 12m^2 + 48 In resulting equation what is the coefficient of the a term? In the resulting equation, what is the constant?

Answers:

Work Shown:

Let's isolate x^2

a = x^2 + 4

x^2 + 4 = a

x^2 = a - 4

Then we'll apply substitution

(x^2+4)^2 + 32 = 12x^2 + 48

a^2 + 32 = 12x^2 + 48 replace x^2+4 with 'a'

a^2 + 32 = 12(a-4) + 48 replace x^2 with a-4

a^2 + 32 = 12a-48+48 distribute

a^2 + 32 = 12a

a^2 + 32 - 12a = 0

a^2 - 12a + 32 = 0

The last equation is in standard form.

The coefficient of the 'a' term is -12 since it is the number to the left of the 'a'. We don't focus on a^2 here.

The constant term is the term without any variable attached, so the constant is 32.

If s₃ is the length of the 3rd side, the possible values are in the interval.

5 < s₃ < 19

Step-by-step explanation:

For a triangle, we know that the sum of any two sides is always larger than the other side.

So if a triangle has 3 sides:

s₁, s₂, and s₃, we have:

s₁ + s₂ > s₃

s₁ + s₃ > s₂

s₂ + s₃ > s₁

In this case, we know that two sides are:

s₁ = 7

s₂ = 12

And we want to find the possible values of s₃.

Then if we use the above inequalities, we get:

7 + 12 > s₃

7 + s₃ > 12

12 + s₃ > 7

With the first one we get:

19 > s₃

Now we can rewrite the other two as:

s₃ > 12 - 7 = 5

s₃ > 7 - 12 = -5

The first one is more restrictive than the second:

s₃ > 5 > -5

So we can use only the first one.

Then the two inequalities are

s₃ > 5

s₃ < 19

Then the range is:

5 < s₃ < 19

This means that any value between 5 and 19 can be a possible length of the third side (5 and 19 are not possible lengths)