0.75x^2+2x=9

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Solution for 0.75x^2+2x=9 equation:


Simplifying
0.75x2 + 2x = 9

Reorder the terms:
2x + 0.75x2 = 9

Solving
2x + 0.75x2 = 9

Solving for variable 'x'.

Reorder the terms:
-9 + 2x + 0.75x2 = 9 + -9

Combine like terms: 9 + -9 = 0
-9 + 2x + 0.75x2 = 0

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

Divide each side by '0.75'.
-12 + 2.666666667x + x2 = 0

Move the constant term to the right:

Add '12' to each side of the equation.
-12 + 2.666666667x + 12 + x2 = 0 + 12

Reorder the terms:
-12 + 12 + 2.666666667x + x2 = 0 + 12

Combine like terms: -12 + 12 = 0
0 + 2.666666667x + x2 = 0 + 12
2.666666667x + x2 = 0 + 12

Combine like terms: 0 + 12 = 12
2.666666667x + x2 = 12

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 = 12 + 1.777777780

Reorder the terms:
1.777777780 + 2.666666667x + x2 = 12 + 1.777777780

Combine like terms: 12 + 1.777777780 = 13.77777778
1.777777780 + 2.666666667x + x2 = 13.77777778

Factor a perfect square on the left side:
(x + 1.333333334)(x + 1.333333334) = 13.77777778

Calculate the square root of the right side: 3.711842909

Break this problem into two subproblems by setting 
(x + 1.333333334) equal to 3.711842909 and -3.711842909.

Subproblem 1

x + 1.333333334 = 3.711842909 Simplifying x + 1.333333334 = 3.711842909 Reorder the terms: 1.333333334 + x = 3.711842909 Solving 1.333333334 + x = 3.711842909 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 = 3.711842909 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + x = 3.711842909 + -1.333333334 x = 3.711842909 + -1.333333334 Combine like terms: 3.711842909 + -1.333333334 = 2.378509575 x = 2.378509575 Simplifying x = 2.378509575

Subproblem 2

x + 1.333333334 = -3.711842909 Simplifying x + 1.333333334 = -3.711842909 Reorder the terms: 1.333333334 + x = -3.711842909 Solving 1.333333334 + x = -3.711842909 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 = -3.711842909 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + x = -3.711842909 + -1.333333334 x = -3.711842909 + -1.333333334 Combine like terms: -3.711842909 + -1.333333334 = -5.045176243 x = -5.045176243 Simplifying x = -5.045176243

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.378509575, -5.045176243}

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