Question

The perimeter of triangle ABC is given by the expression 3x2 - 7x + 2.

if AB = 2x - 10x +6 and BC = x-x-4. write an expression in simplest form and standard
form to represent side AC.

Answer 1

Perimeter of ABC = AB + BC+AC. Replace letters with equations.

3x^2-7x+2 = 2x^2-10x+6 + x^2-x-4 + AC = 3x^2-11x+2 + AC.

AC = 4x;

Answer 2

Question 1:

For this case we must simplify the following expression:

`(6k-k ^ 2 + 6) - (5 + 7k + 6k ^ 2) - (1 + 3k ^ 2-4k)`

We eliminate the parentheses taking into account that:

`- * + = -\\- * - = +\\6k-k ^ 2 + 6-5-7k-6k ^ 2-1-3k ^ 2 + 4k =`

We group similar terms:

`-k ^ 2-6k ^ 2-3k ^ 2 + 6k-7k + 4k + 6-5-1 =`

We take into account that:

Same signs add up and the same sign is placed

Different signs are subtracted and the major sign is placed

`-10k ^ 2 + 3k + 0 =\\-10k ^ 2 + 3k`

Answer:

`-10k ^ 2 + 3k`

Question 2:

For this case we must simplify the following expression:

`(5a^4-2a^2-6a^3)+(2a-3a^3-a^4)-(7a+6a^4-5a^3)`

We eliminate the parentheses taking into account that:

`- * + = -\\- * - = +\\5a^4-2a^2-6a^3+2a-3a^3-a^4-7a-6a^4+5a^3=`

We group similar terms:

`5a^4-a^4-6a^4-6a^3-3a^3+5a^3-2a^2+2a-7a=`

We take into account that:

Same signs add up and the same sign is placed

Different signs are subtracted and the major sign is placed

`-2a ^ 4-4a ^ 3-2a ^ 2-5a`

Answer:

`-2a ^ 4-4a ^ 3-2a ^ 2-5a`

Question 3:

For this case we must simplify the following expression:

`(3k ^ 4 + k ^ 2-5 + 3k) + (1-5k + 4k ^ 4-8k ^ 3) + (5 + k ^ 2-3k ^ 4 + 3k)`

We eliminate the parentheses taking into account that:

`- * + = -\\- * - = +\\3k ^ 4 + k ^ 2-5 + 3k + 1-5k + 4k ^ 4-8k ^ 3 + 5 + k ^ 2-3k ^ 4 + 3k =`

We group similar terms:

`3k ^ 4 + 4k ^ 4-3k ^ 4-8k ^ 3 + k ^ 2 + k ^ 2 + 3k-5k + 3k-5 + 1 + 5 =`

We take into account that:

Same signs add up and the same sign is placed

Different signs are subtracted and the major sign is placed

`4k ^ 4-8k ^ 3 + 2k ^ 2 + k + 1`

ANswer:

`4k ^ 4-8k ^ 3 + 2k ^ 2 + k + 1`

Question 4:

For this case we must simplify the following expression:

`(4v-7-7v ^ 2-5v ^ 4) + (2v ^ 3-4 + 2v-8v ^ 2) + (8v-6v ^ 3-1 + 4v ^ 2)`

We eliminate the parentheses taking into account that:

`- * + = -\\- * - = +\\4v-7-7v ^ 2-5v ^ 4 + 2v ^ 3-4 + 2v-8v ^ 2 + 8v-6v ^ 3-1 + 4v ^ 2 =`

We group similar terms:

`-5v ^ 4 + 2v ^ 3-6v ^ 3-7v ^ 2-8v ^ 2 + 4v ^ 2 + 4v + 2v + 8v-7-4-1 =`

We take into account that:

Same signs add up and the same sign is placed

Different signs are subtracted and the major sign is placed

`-5v ^ 4-4v ^ 3-11v ^ 2 + 14v-12`

Answer:

`-5v ^ 4-4v ^ 3-11v ^ 2 + 14v-12`

Question 5:

For this case we have that by definition, the perimeter of the triangle will be given by the sum of its sides. That is to say:

`3x ^ 2-7x + 2 = (2x ^ 2-10x + 6) + (x ^ 2-x-4) + AC`

We simplify the right side of the equation:

`3x ^ 2-7x + 2 = 2x ^ 2-10x + 6 + x ^ 2-x-4 + AC\\3x ^ 2-7x + 2 = 3x ^ 2-11x + 4 + AC`

We cleared AC:

`AC = 3x ^ 2-7x + 2- (3x ^ 2-11x + 4)\\- * + = -\\- * - = +\\AC = 3x ^ 2-7x + 2-3x ^ 2 + 11x-4\\AC = 4x-2`

Answer:

`AC = 4x-2`