3x^2+14x=45

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Solution for 3x^2+14x=45 equation:


Simplifying
3x2 + 14x = 45

Reorder the terms:
14x + 3x2 = 45

Solving
14x + 3x2 = 45

Solving for variable 'x'.

Reorder the terms:
-45 + 14x + 3x2 = 45 + -45

Combine like terms: 45 + -45 = 0
-45 + 14x + 3x2 = 0

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

Divide each side by '3'.
-15 + 4.666666667x + x2 = 0

Move the constant term to the right:

Add '15' to each side of the equation.
-15 + 4.666666667x + 15 + x2 = 0 + 15

Reorder the terms:
-15 + 15 + 4.666666667x + x2 = 0 + 15

Combine like terms: -15 + 15 = 0
0 + 4.666666667x + x2 = 0 + 15
4.666666667x + x2 = 0 + 15

Combine like terms: 0 + 15 = 15
4.666666667x + x2 = 15

The x term is 4.666666667x.  Take half its coefficient (2.333333334).
Square it (5.444444448) and add it to both sides.

Add '5.444444448' to each side of the equation.
4.666666667x + 5.444444448 + x2 = 15 + 5.444444448

Reorder the terms:
5.444444448 + 4.666666667x + x2 = 15 + 5.444444448

Combine like terms: 15 + 5.444444448 = 20.444444448
5.444444448 + 4.666666667x + x2 = 20.444444448

Factor a perfect square on the left side:
(x + 2.333333334)(x + 2.333333334) = 20.444444448

Calculate the square root of the right side: 4.521553322

Break this problem into two subproblems by setting 
(x + 2.333333334) equal to 4.521553322 and -4.521553322.

Subproblem 1

x + 2.333333334 = 4.521553322 Simplifying x + 2.333333334 = 4.521553322 Reorder the terms: 2.333333334 + x = 4.521553322 Solving 2.333333334 + x = 4.521553322 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.333333334' to each side of the equation. 2.333333334 + -2.333333334 + x = 4.521553322 + -2.333333334 Combine like terms: 2.333333334 + -2.333333334 = 0.000000000 0.000000000 + x = 4.521553322 + -2.333333334 x = 4.521553322 + -2.333333334 Combine like terms: 4.521553322 + -2.333333334 = 2.188219988 x = 2.188219988 Simplifying x = 2.188219988

Subproblem 2

x + 2.333333334 = -4.521553322 Simplifying x + 2.333333334 = -4.521553322 Reorder the terms: 2.333333334 + x = -4.521553322 Solving 2.333333334 + x = -4.521553322 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.333333334' to each side of the equation. 2.333333334 + -2.333333334 + x = -4.521553322 + -2.333333334 Combine like terms: 2.333333334 + -2.333333334 = 0.000000000 0.000000000 + x = -4.521553322 + -2.333333334 x = -4.521553322 + -2.333333334 Combine like terms: -4.521553322 + -2.333333334 = -6.854886656 x = -6.854886656 Simplifying x = -6.854886656

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.188219988, -6.854886656}

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