4x^2=5x+10

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Solution for 4x^2=5x+10 equation:


Simplifying
4x2 = 5x + 10

Reorder the terms:
4x2 = 10 + 5x

Solving
4x2 = 10 + 5x

Solving for variable 'x'.

Reorder the terms:
-10 + -5x + 4x2 = 10 + 5x + -10 + -5x

Reorder the terms:
-10 + -5x + 4x2 = 10 + -10 + 5x + -5x

Combine like terms: 10 + -10 = 0
-10 + -5x + 4x2 = 0 + 5x + -5x
-10 + -5x + 4x2 = 5x + -5x

Combine like terms: 5x + -5x = 0
-10 + -5x + 4x2 = 0

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

Divide each side by '4'.
-2.5 + -1.25x + x2 = 0

Move the constant term to the right:

Add '2.5' to each side of the equation.
-2.5 + -1.25x + 2.5 + x2 = 0 + 2.5

Reorder the terms:
-2.5 + 2.5 + -1.25x + x2 = 0 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + -1.25x + x2 = 0 + 2.5
-1.25x + x2 = 0 + 2.5

Combine like terms: 0 + 2.5 = 2.5
-1.25x + x2 = 2.5

The x term is -1.25x.  Take half its coefficient (-0.625).
Square it (0.390625) and add it to both sides.

Add '0.390625' to each side of the equation.
-1.25x + 0.390625 + x2 = 2.5 + 0.390625

Reorder the terms:
0.390625 + -1.25x + x2 = 2.5 + 0.390625

Combine like terms: 2.5 + 0.390625 = 2.890625
0.390625 + -1.25x + x2 = 2.890625

Factor a perfect square on the left side:
(x + -0.625)(x + -0.625) = 2.890625

Calculate the square root of the right side: 1.700183814

Break this problem into two subproblems by setting 
(x + -0.625) equal to 1.700183814 and -1.700183814.

Subproblem 1

x + -0.625 = 1.700183814 Simplifying x + -0.625 = 1.700183814 Reorder the terms: -0.625 + x = 1.700183814 Solving -0.625 + x = 1.700183814 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + x = 1.700183814 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + x = 1.700183814 + 0.625 x = 1.700183814 + 0.625 Combine like terms: 1.700183814 + 0.625 = 2.325183814 x = 2.325183814 Simplifying x = 2.325183814

Subproblem 2

x + -0.625 = -1.700183814 Simplifying x + -0.625 = -1.700183814 Reorder the terms: -0.625 + x = -1.700183814 Solving -0.625 + x = -1.700183814 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + x = -1.700183814 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + x = -1.700183814 + 0.625 x = -1.700183814 + 0.625 Combine like terms: -1.700183814 + 0.625 = -1.075183814 x = -1.075183814 Simplifying x = -1.075183814

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.325183814, -1.075183814}

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