5x^2+x-10.5=0

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Solution for 5x^2+x-10.5=0 equation:


Simplifying
5x2 + x + -10.5 = 0

Reorder the terms:
-10.5 + x + 5x2 = 0

Solving
-10.5 + x + 5x2 = 0

Solving for variable 'x'.

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

Divide each side by '5'.
-2.1 + 0.2x + x2 = 0

Move the constant term to the right:

Add '2.1' to each side of the equation.
-2.1 + 0.2x + 2.1 + x2 = 0 + 2.1

Reorder the terms:
-2.1 + 2.1 + 0.2x + x2 = 0 + 2.1

Combine like terms: -2.1 + 2.1 = 0.0
0.0 + 0.2x + x2 = 0 + 2.1
0.2x + x2 = 0 + 2.1

Combine like terms: 0 + 2.1 = 2.1
0.2x + x2 = 2.1

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.2x + 0.25 + x2 = 2.1 + 0.25

Reorder the terms:
0.25 + 0.2x + x2 = 2.1 + 0.25

Combine like terms: 2.1 + 0.25 = 2.35
0.25 + 0.2x + x2 = 2.35

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 2.35

Calculate the square root of the right side: 1.532970972

Break this problem into two subproblems by setting 
(x + 0.5) equal to 1.532970972 and -1.532970972.

Subproblem 1

x + 0.5 = 1.532970972 Simplifying x + 0.5 = 1.532970972 Reorder the terms: 0.5 + x = 1.532970972 Solving 0.5 + x = 1.532970972 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 1.532970972 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 1.532970972 + -0.5 x = 1.532970972 + -0.5 Combine like terms: 1.532970972 + -0.5 = 1.032970972 x = 1.032970972 Simplifying x = 1.032970972

Subproblem 2

x + 0.5 = -1.532970972 Simplifying x + 0.5 = -1.532970972 Reorder the terms: 0.5 + x = -1.532970972 Solving 0.5 + x = -1.532970972 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -1.532970972 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -1.532970972 + -0.5 x = -1.532970972 + -0.5 Combine like terms: -1.532970972 + -0.5 = -2.032970972 x = -2.032970972 Simplifying x = -2.032970972

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.032970972, -2.032970972}

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