4x^2=7x+13

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Solution for 4x^2=7x+13 equation:


Simplifying
4x2 = 7x + 13

Reorder the terms:
4x2 = 13 + 7x

Solving
4x2 = 13 + 7x

Solving for variable 'x'.

Reorder the terms:
-13 + -7x + 4x2 = 13 + 7x + -13 + -7x

Reorder the terms:
-13 + -7x + 4x2 = 13 + -13 + 7x + -7x

Combine like terms: 13 + -13 = 0
-13 + -7x + 4x2 = 0 + 7x + -7x
-13 + -7x + 4x2 = 7x + -7x

Combine like terms: 7x + -7x = 0
-13 + -7x + 4x2 = 0

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

Divide each side by '4'.
-3.25 + -1.75x + x2 = 0

Move the constant term to the right:

Add '3.25' to each side of the equation.
-3.25 + -1.75x + 3.25 + x2 = 0 + 3.25

Reorder the terms:
-3.25 + 3.25 + -1.75x + x2 = 0 + 3.25

Combine like terms: -3.25 + 3.25 = 0.00
0.00 + -1.75x + x2 = 0 + 3.25
-1.75x + x2 = 0 + 3.25

Combine like terms: 0 + 3.25 = 3.25
-1.75x + x2 = 3.25

The x term is -1.75x.  Take half its coefficient (-0.875).
Square it (0.765625) and add it to both sides.

Add '0.765625' to each side of the equation.
-1.75x + 0.765625 + x2 = 3.25 + 0.765625

Reorder the terms:
0.765625 + -1.75x + x2 = 3.25 + 0.765625

Combine like terms: 3.25 + 0.765625 = 4.015625
0.765625 + -1.75x + x2 = 4.015625

Factor a perfect square on the left side:
(x + -0.875)(x + -0.875) = 4.015625

Calculate the square root of the right side: 2.003902443

Break this problem into two subproblems by setting 
(x + -0.875) equal to 2.003902443 and -2.003902443.

Subproblem 1

x + -0.875 = 2.003902443 Simplifying x + -0.875 = 2.003902443 Reorder the terms: -0.875 + x = 2.003902443 Solving -0.875 + x = 2.003902443 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + x = 2.003902443 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + x = 2.003902443 + 0.875 x = 2.003902443 + 0.875 Combine like terms: 2.003902443 + 0.875 = 2.878902443 x = 2.878902443 Simplifying x = 2.878902443

Subproblem 2

x + -0.875 = -2.003902443 Simplifying x + -0.875 = -2.003902443 Reorder the terms: -0.875 + x = -2.003902443 Solving -0.875 + x = -2.003902443 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + x = -2.003902443 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + x = -2.003902443 + 0.875 x = -2.003902443 + 0.875 Combine like terms: -2.003902443 + 0.875 = -1.128902443 x = -1.128902443 Simplifying x = -1.128902443

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.878902443, -1.128902443}

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