7.5x^2-3x-10.5=0

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Solution for 7.5x^2-3x-10.5=0 equation:


Simplifying
7.5x2 + -3x + -10.5 = 0

Reorder the terms:
-10.5 + -3x + 7.5x2 = 0

Solving
-10.5 + -3x + 7.5x2 = 0

Solving for variable 'x'.

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

Divide each side by '7.5'.
-1.4 + -0.4x + x2 = 0

Move the constant term to the right:

Add '1.4' to each side of the equation.
-1.4 + -0.4x + 1.4 + x2 = 0 + 1.4

Reorder the terms:
-1.4 + 1.4 + -0.4x + x2 = 0 + 1.4

Combine like terms: -1.4 + 1.4 = 0.0
0.0 + -0.4x + x2 = 0 + 1.4
-0.4x + x2 = 0 + 1.4

Combine like terms: 0 + 1.4 = 1.4
-0.4x + x2 = 1.4

The x term is -0.4x.  Take half its coefficient (-0.2).
Square it (0.04) and add it to both sides.

Add '0.04' to each side of the equation.
-0.4x + 0.04 + x2 = 1.4 + 0.04

Reorder the terms:
0.04 + -0.4x + x2 = 1.4 + 0.04

Combine like terms: 1.4 + 0.04 = 1.44
0.04 + -0.4x + x2 = 1.44

Factor a perfect square on the left side:
(x + -0.2)(x + -0.2) = 1.44

Calculate the square root of the right side: 1.2

Break this problem into two subproblems by setting 
(x + -0.2) equal to 1.2 and -1.2.

Subproblem 1

x + -0.2 = 1.2 Simplifying x + -0.2 = 1.2 Reorder the terms: -0.2 + x = 1.2 Solving -0.2 + x = 1.2 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.2' to each side of the equation. -0.2 + 0.2 + x = 1.2 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + x = 1.2 + 0.2 x = 1.2 + 0.2 Combine like terms: 1.2 + 0.2 = 1.4 x = 1.4 Simplifying x = 1.4

Subproblem 2

x + -0.2 = -1.2 Simplifying x + -0.2 = -1.2 Reorder the terms: -0.2 + x = -1.2 Solving -0.2 + x = -1.2 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.2' to each side of the equation. -0.2 + 0.2 + x = -1.2 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + x = -1.2 + 0.2 x = -1.2 + 0.2 Combine like terms: -1.2 + 0.2 = -1 x = -1 Simplifying x = -1

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.4, -1}

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