9x^2+24x+16=7

Solution for 9x^2+24x+16=7 equation:

Simplifying
9x2 + 24x + 16 = 7

Reorder the terms:
16 + 24x + 9x2 = 7

Solving
16 + 24x + 9x2 = 7

Solving for variable 'x'.

Reorder the terms:
16 + -7 + 24x + 9x2 = 7 + -7

Combine like terms: 16 + -7 = 9
9 + 24x + 9x2 = 7 + -7

Combine like terms: 7 + -7 = 0
9 + 24x + 9x2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(3 + 8x + 3x2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(3 + 8x + 3x2)' equal to zero and attempt to solve:

Simplifying
3 + 8x + 3x2 = 0

Solving
3 + 8x + 3x2 = 0

Begin completing the square.  Divide all terms by
3 the coefficient of the squared term: 

Divide each side by '3'.
1 + 2.666666667x + x2 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 2.666666667x + -1 + x2 = 0 + -1

Reorder the terms:
1 + -1 + 2.666666667x + x2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 2.666666667x + x2 = 0 + -1
2.666666667x + x2 = 0 + -1

Combine like terms: 0 + -1 = -1
2.666666667x + x2 = -1

The x term is 2.666666667x.  Take half its coefficient (1.333333334).
Square it (1.777777780) and add it to both sides.

Add '1.777777780' to each side of the equation.
2.666666667x + 1.777777780 + x2 = -1 + 1.777777780

Reorder the terms:
1.777777780 + 2.666666667x + x2 = -1 + 1.777777780

Combine like terms: -1 + 1.777777780 = 0.77777778
1.777777780 + 2.666666667x + x2 = 0.77777778

Factor a perfect square on the left side:
(x + 1.333333334)(x + 1.333333334) = 0.77777778

Calculate the square root of the right side: 0.881917105

Break this problem into two subproblems by setting 
(x + 1.333333334) equal to 0.881917105 and -0.881917105.

Subproblem 1
x + 1.333333334 = 0.881917105

Simplifying
x + 1.333333334 = 0.881917105

Reorder the terms:
1.333333334 + x = 0.881917105

Solving
1.333333334 + x = 0.881917105

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-1.333333334' to each side of the equation.
1.333333334 + -1.333333334 + x = 0.881917105 + -1.333333334

Combine like terms: 1.333333334 + -1.333333334 = 0.000000000
0.000000000 + x = 0.881917105 + -1.333333334
x = 0.881917105 + -1.333333334

Combine like terms: 0.881917105 + -1.333333334 = -0.451416229
x = -0.451416229

Simplifying
x = -0.451416229

Subproblem 2
x + 1.333333334 = -0.881917105

Simplifying
x + 1.333333334 = -0.881917105

Reorder the terms:
1.333333334 + x = -0.881917105

Solving
1.333333334 + x = -0.881917105

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-1.333333334' to each side of the equation.
1.333333334 + -1.333333334 + x = -0.881917105 + -1.333333334

Combine like terms: 1.333333334 + -1.333333334 = 0.000000000
0.000000000 + x = -0.881917105 + -1.333333334
x = -0.881917105 + -1.333333334

Combine like terms: -0.881917105 + -1.333333334 = -2.215250439
x = -2.215250439

Simplifying
x = -2.215250439

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.451416229, -2.215250439}
Solution
x = {-0.451416229, -2.215250439}