4x^2=3x+16

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Solution for 4x^2=3x+16 equation:


Simplifying
4x2 = 3x + 16

Reorder the terms:
4x2 = 16 + 3x

Solving
4x2 = 16 + 3x

Solving for variable 'x'.

Reorder the terms:
-16 + -3x + 4x2 = 16 + 3x + -16 + -3x

Reorder the terms:
-16 + -3x + 4x2 = 16 + -16 + 3x + -3x

Combine like terms: 16 + -16 = 0
-16 + -3x + 4x2 = 0 + 3x + -3x
-16 + -3x + 4x2 = 3x + -3x

Combine like terms: 3x + -3x = 0
-16 + -3x + 4x2 = 0

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

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

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + -0.75x + 4 + x2 = 0 + 4

Reorder the terms:
-4 + 4 + -0.75x + x2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + -0.75x + x2 = 0 + 4
-0.75x + x2 = 0 + 4

Combine like terms: 0 + 4 = 4
-0.75x + x2 = 4

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 = 4 + 0.140625

Reorder the terms:
0.140625 + -0.75x + x2 = 4 + 0.140625

Combine like terms: 4 + 0.140625 = 4.140625
0.140625 + -0.75x + x2 = 4.140625

Factor a perfect square on the left side:
(x + -0.375)(x + -0.375) = 4.140625

Calculate the square root of the right side: 2.034852575

Break this problem into two subproblems by setting 
(x + -0.375) equal to 2.034852575 and -2.034852575.

Subproblem 1

x + -0.375 = 2.034852575 Simplifying x + -0.375 = 2.034852575 Reorder the terms: -0.375 + x = 2.034852575 Solving -0.375 + x = 2.034852575 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 = 2.034852575 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + x = 2.034852575 + 0.375 x = 2.034852575 + 0.375 Combine like terms: 2.034852575 + 0.375 = 2.409852575 x = 2.409852575 Simplifying x = 2.409852575

Subproblem 2

x + -0.375 = -2.034852575 Simplifying x + -0.375 = -2.034852575 Reorder the terms: -0.375 + x = -2.034852575 Solving -0.375 + x = -2.034852575 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 = -2.034852575 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + x = -2.034852575 + 0.375 x = -2.034852575 + 0.375 Combine like terms: -2.034852575 + 0.375 = -1.659852575 x = -1.659852575 Simplifying x = -1.659852575

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.409852575, -1.659852575}

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