4x^2=3x+9

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Solution for 4x^2=3x+9 equation:


Simplifying
4x2 = 3x + 9

Reorder the terms:
4x2 = 9 + 3x

Solving
4x2 = 9 + 3x

Solving for variable 'x'.

Reorder the terms:
-9 + -3x + 4x2 = 9 + 3x + -9 + -3x

Reorder the terms:
-9 + -3x + 4x2 = 9 + -9 + 3x + -3x

Combine like terms: 9 + -9 = 0
-9 + -3x + 4x2 = 0 + 3x + -3x
-9 + -3x + 4x2 = 3x + -3x

Combine like terms: 3x + -3x = 0
-9 + -3x + 4x2 = 0

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

Divide each side by '4'.
-2.25 + -0.75x + x2 = 0

Move the constant term to the right:

Add '2.25' to each side of the equation.
-2.25 + -0.75x + 2.25 + x2 = 0 + 2.25

Reorder the terms:
-2.25 + 2.25 + -0.75x + x2 = 0 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + -0.75x + x2 = 0 + 2.25
-0.75x + x2 = 0 + 2.25

Combine like terms: 0 + 2.25 = 2.25
-0.75x + x2 = 2.25

The x term is -0.75x.  Take half its coefficient (-0.375).
Square it (0.140625) and add it to both sides.

Add '0.140625' to each side of the equation.
-0.75x + 0.140625 + x2 = 2.25 + 0.140625

Reorder the terms:
0.140625 + -0.75x + x2 = 2.25 + 0.140625

Combine like terms: 2.25 + 0.140625 = 2.390625
0.140625 + -0.75x + x2 = 2.390625

Factor a perfect square on the left side:
(x + -0.375)(x + -0.375) = 2.390625

Calculate the square root of the right side: 1.54616461

Break this problem into two subproblems by setting 
(x + -0.375) equal to 1.54616461 and -1.54616461.

Subproblem 1

x + -0.375 = 1.54616461 Simplifying x + -0.375 = 1.54616461 Reorder the terms: -0.375 + x = 1.54616461 Solving -0.375 + x = 1.54616461 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + x = 1.54616461 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + x = 1.54616461 + 0.375 x = 1.54616461 + 0.375 Combine like terms: 1.54616461 + 0.375 = 1.92116461 x = 1.92116461 Simplifying x = 1.92116461

Subproblem 2

x + -0.375 = -1.54616461 Simplifying x + -0.375 = -1.54616461 Reorder the terms: -0.375 + x = -1.54616461 Solving -0.375 + x = -1.54616461 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + x = -1.54616461 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + x = -1.54616461 + 0.375 x = -1.54616461 + 0.375 Combine like terms: -1.54616461 + 0.375 = -1.17116461 x = -1.17116461 Simplifying x = -1.17116461

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.92116461, -1.17116461}

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