7.5x^2-30x+30=x^2

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Solution for 7.5x^2-30x+30=x^2 equation:


Simplifying
7.5x2 + -30x + 30 = x2

Reorder the terms:
30 + -30x + 7.5x2 = x2

Solving
30 + -30x + 7.5x2 = x2

Solving for variable 'x'.

Combine like terms: 7.5x2 + -1x2 = 6.5x2
30 + -30x + 6.5x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
30 + -30x + 6.5x2 = 0

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

Divide each side by '6.5'.
4.615384615 + -4.615384615x + x2 = 0

Move the constant term to the right:

Add '-4.615384615' to each side of the equation.
4.615384615 + -4.615384615x + -4.615384615 + x2 = 0 + -4.615384615

Reorder the terms:
4.615384615 + -4.615384615 + -4.615384615x + x2 = 0 + -4.615384615

Combine like terms: 4.615384615 + -4.615384615 = 0.000000000
0.000000000 + -4.615384615x + x2 = 0 + -4.615384615
-4.615384615x + x2 = 0 + -4.615384615

Combine like terms: 0 + -4.615384615 = -4.615384615
-4.615384615x + x2 = -4.615384615

The x term is -4.615384615x.  Take half its coefficient (-2.307692308).
Square it (5.325443788) and add it to both sides.

Add '5.325443788' to each side of the equation.
-4.615384615x + 5.325443788 + x2 = -4.615384615 + 5.325443788

Reorder the terms:
5.325443788 + -4.615384615x + x2 = -4.615384615 + 5.325443788

Combine like terms: -4.615384615 + 5.325443788 = 0.710059173
5.325443788 + -4.615384615x + x2 = 0.710059173

Factor a perfect square on the left side:
(x + -2.307692308)(x + -2.307692308) = 0.710059173

Calculate the square root of the right side: 0.842650089

Break this problem into two subproblems by setting 
(x + -2.307692308) equal to 0.842650089 and -0.842650089.

Subproblem 1

x + -2.307692308 = 0.842650089 Simplifying x + -2.307692308 = 0.842650089 Reorder the terms: -2.307692308 + x = 0.842650089 Solving -2.307692308 + x = 0.842650089 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.307692308' to each side of the equation. -2.307692308 + 2.307692308 + x = 0.842650089 + 2.307692308 Combine like terms: -2.307692308 + 2.307692308 = 0.000000000 0.000000000 + x = 0.842650089 + 2.307692308 x = 0.842650089 + 2.307692308 Combine like terms: 0.842650089 + 2.307692308 = 3.150342397 x = 3.150342397 Simplifying x = 3.150342397

Subproblem 2

x + -2.307692308 = -0.842650089 Simplifying x + -2.307692308 = -0.842650089 Reorder the terms: -2.307692308 + x = -0.842650089 Solving -2.307692308 + x = -0.842650089 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.307692308' to each side of the equation. -2.307692308 + 2.307692308 + x = -0.842650089 + 2.307692308 Combine like terms: -2.307692308 + 2.307692308 = 0.000000000 0.000000000 + x = -0.842650089 + 2.307692308 x = -0.842650089 + 2.307692308 Combine like terms: -0.842650089 + 2.307692308 = 1.465042219 x = 1.465042219 Simplifying x = 1.465042219

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.150342397, 1.465042219}

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