The sum of the angles of a triangle is 180 degrees. So, this means:

`3x + (x^2+1) + (x^2+6) = 180`

Combine like terms: `2x^2 + 3x + 7 = 180`

In order to solve a quadratic we could factor, complete the square, or use quadratic formula.

Solve: `2x^2 + 3x - 173 = 0`

`(-3+-sqrt((3^2) - 4(2)(-173)))/(2(2))`

` `

` ` `(-3+-sqrt(1393))/4`

` `Therefore `x = (-3+-sqrt(1393))/4`

In decimal equivalency `x ~~ -10.081or x~~ 8.581`

Since we are solving for x which represents part of an angle we can not have a negative angle measure therefore:

The final answer is:`x = (-3+sqrt(1393))/4` or `x~~ 8.581`

The sum of the angles of a triangle is equal to 180 degrees. If the three angles of a triangle are 3x, x^2 + 1 and x^2 + 6, to determine x, solve the equation:

3x + x^2 + 1 + x^2 + 6 = 180

=> 2x^2 + 3x + 7 = 180

=> 2x^2 + 3x - 173 = 0

=> x = `(-3+-sqrt(9 + 1384))/4`

=> x = `-3/4 +- sqrt 1393/4`

The negative root can be eliminated as the angle of a triangle cannot be negative.

**This gives **`x = -3/4 +sqrt 1393/4`

3x + x^2 + 1 + x^2 + 6 = 180

combine like terms

2x^2 + 3x + 7 = 180

move the 180 to the other side by subtracting it.

2x^2 + 3x - 173 = 0

now plug it into the quadratic formula:

a = 2 b = 3 c = -173

`(-3 +-sqrt(3^2 - 4(2)(-173)))/(2(2))`

simplify:

`(-3 +-sqrt(9 + 1384))/(4)`

`(-3 +-sqrt(1393))/(4)`

`x =(-3 +-sqrt(1393))/(4)`

`x = (-3 +37.3)/4`

`x = 34.3 / 4`

` `

**x= 8.6**

`x = (-3 -37.3)/4`

`x = -40.3 / 4`

**x= -10**