6x^2=9x+7

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Solution for 6x^2=9x+7 equation:


Simplifying
6x2 = 9x + 7

Reorder the terms:
6x2 = 7 + 9x

Solving
6x2 = 7 + 9x

Solving for variable 'x'.

Reorder the terms:
-7 + -9x + 6x2 = 7 + 9x + -7 + -9x

Reorder the terms:
-7 + -9x + 6x2 = 7 + -7 + 9x + -9x

Combine like terms: 7 + -7 = 0
-7 + -9x + 6x2 = 0 + 9x + -9x
-7 + -9x + 6x2 = 9x + -9x

Combine like terms: 9x + -9x = 0
-7 + -9x + 6x2 = 0

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

Divide each side by '6'.
-1.166666667 + -1.5x + x2 = 0

Move the constant term to the right:

Add '1.166666667' to each side of the equation.
-1.166666667 + -1.5x + 1.166666667 + x2 = 0 + 1.166666667

Reorder the terms:
-1.166666667 + 1.166666667 + -1.5x + x2 = 0 + 1.166666667

Combine like terms: -1.166666667 + 1.166666667 = 0.000000000
0.000000000 + -1.5x + x2 = 0 + 1.166666667
-1.5x + x2 = 0 + 1.166666667

Combine like terms: 0 + 1.166666667 = 1.166666667
-1.5x + x2 = 1.166666667

The x term is -1.5x.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5x + 0.5625 + x2 = 1.166666667 + 0.5625

Reorder the terms:
0.5625 + -1.5x + x2 = 1.166666667 + 0.5625

Combine like terms: 1.166666667 + 0.5625 = 1.729166667
0.5625 + -1.5x + x2 = 1.729166667

Factor a perfect square on the left side:
(x + -0.75)(x + -0.75) = 1.729166667

Calculate the square root of the right side: 1.31497782

Break this problem into two subproblems by setting 
(x + -0.75) equal to 1.31497782 and -1.31497782.

Subproblem 1

x + -0.75 = 1.31497782 Simplifying x + -0.75 = 1.31497782 Reorder the terms: -0.75 + x = 1.31497782 Solving -0.75 + x = 1.31497782 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = 1.31497782 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = 1.31497782 + 0.75 x = 1.31497782 + 0.75 Combine like terms: 1.31497782 + 0.75 = 2.06497782 x = 2.06497782 Simplifying x = 2.06497782

Subproblem 2

x + -0.75 = -1.31497782 Simplifying x + -0.75 = -1.31497782 Reorder the terms: -0.75 + x = -1.31497782 Solving -0.75 + x = -1.31497782 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = -1.31497782 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = -1.31497782 + 0.75 x = -1.31497782 + 0.75 Combine like terms: -1.31497782 + 0.75 = -0.56497782 x = -0.56497782 Simplifying x = -0.56497782

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.06497782, -0.56497782}

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