10x^2=27x-16

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Solution for 10x^2=27x-16 equation:


Simplifying
10x2 = 27x + -16

Reorder the terms:
10x2 = -16 + 27x

Solving
10x2 = -16 + 27x

Solving for variable 'x'.

Reorder the terms:
16 + -27x + 10x2 = -16 + 27x + 16 + -27x

Reorder the terms:
16 + -27x + 10x2 = -16 + 16 + 27x + -27x

Combine like terms: -16 + 16 = 0
16 + -27x + 10x2 = 0 + 27x + -27x
16 + -27x + 10x2 = 27x + -27x

Combine like terms: 27x + -27x = 0
16 + -27x + 10x2 = 0

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

Divide each side by '10'.
1.6 + -2.7x + x2 = 0

Move the constant term to the right:

Add '-1.6' to each side of the equation.
1.6 + -2.7x + -1.6 + x2 = 0 + -1.6

Reorder the terms:
1.6 + -1.6 + -2.7x + x2 = 0 + -1.6

Combine like terms: 1.6 + -1.6 = 0.0
0.0 + -2.7x + x2 = 0 + -1.6
-2.7x + x2 = 0 + -1.6

Combine like terms: 0 + -1.6 = -1.6
-2.7x + x2 = -1.6

The x term is -2.7x.  Take half its coefficient (-1.35).
Square it (1.8225) and add it to both sides.

Add '1.8225' to each side of the equation.
-2.7x + 1.8225 + x2 = -1.6 + 1.8225

Reorder the terms:
1.8225 + -2.7x + x2 = -1.6 + 1.8225

Combine like terms: -1.6 + 1.8225 = 0.2225
1.8225 + -2.7x + x2 = 0.2225

Factor a perfect square on the left side:
(x + -1.35)(x + -1.35) = 0.2225

Calculate the square root of the right side: 0.471699057

Break this problem into two subproblems by setting 
(x + -1.35) equal to 0.471699057 and -0.471699057.

Subproblem 1

x + -1.35 = 0.471699057 Simplifying x + -1.35 = 0.471699057 Reorder the terms: -1.35 + x = 0.471699057 Solving -1.35 + x = 0.471699057 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.35' to each side of the equation. -1.35 + 1.35 + x = 0.471699057 + 1.35 Combine like terms: -1.35 + 1.35 = 0.00 0.00 + x = 0.471699057 + 1.35 x = 0.471699057 + 1.35 Combine like terms: 0.471699057 + 1.35 = 1.821699057 x = 1.821699057 Simplifying x = 1.821699057

Subproblem 2

x + -1.35 = -0.471699057 Simplifying x + -1.35 = -0.471699057 Reorder the terms: -1.35 + x = -0.471699057 Solving -1.35 + x = -0.471699057 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.35' to each side of the equation. -1.35 + 1.35 + x = -0.471699057 + 1.35 Combine like terms: -1.35 + 1.35 = 0.00 0.00 + x = -0.471699057 + 1.35 x = -0.471699057 + 1.35 Combine like terms: -0.471699057 + 1.35 = 0.878300943 x = 0.878300943 Simplifying x = 0.878300943

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.821699057, 0.878300943}

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