4x^2=7x+7.5

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Solution for 4x^2=7x+7.5 equation:


Simplifying
4x2 = 7x + 7.5

Reorder the terms:
4x2 = 7.5 + 7x

Solving
4x2 = 7.5 + 7x

Solving for variable 'x'.

Reorder the terms:
-7.5 + -7x + 4x2 = 7.5 + 7x + -7.5 + -7x

Reorder the terms:
-7.5 + -7x + 4x2 = 7.5 + -7.5 + 7x + -7x

Combine like terms: 7.5 + -7.5 = 0.0
-7.5 + -7x + 4x2 = 0.0 + 7x + -7x
-7.5 + -7x + 4x2 = 7x + -7x

Combine like terms: 7x + -7x = 0
-7.5 + -7x + 4x2 = 0

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

Divide each side by '4'.
-1.875 + -1.75x + x2 = 0

Move the constant term to the right:

Add '1.875' to each side of the equation.
-1.875 + -1.75x + 1.875 + x2 = 0 + 1.875

Reorder the terms:
-1.875 + 1.875 + -1.75x + x2 = 0 + 1.875

Combine like terms: -1.875 + 1.875 = 0.000
0.000 + -1.75x + x2 = 0 + 1.875
-1.75x + x2 = 0 + 1.875

Combine like terms: 0 + 1.875 = 1.875
-1.75x + x2 = 1.875

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 = 1.875 + 0.765625

Reorder the terms:
0.765625 + -1.75x + x2 = 1.875 + 0.765625

Combine like terms: 1.875 + 0.765625 = 2.640625
0.765625 + -1.75x + x2 = 2.640625

Factor a perfect square on the left side:
(x + -0.875)(x + -0.875) = 2.640625

Calculate the square root of the right side: 1.625

Break this problem into two subproblems by setting 
(x + -0.875) equal to 1.625 and -1.625.

Subproblem 1

x + -0.875 = 1.625 Simplifying x + -0.875 = 1.625 Reorder the terms: -0.875 + x = 1.625 Solving -0.875 + x = 1.625 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 = 1.625 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + x = 1.625 + 0.875 x = 1.625 + 0.875 Combine like terms: 1.625 + 0.875 = 2.5 x = 2.5 Simplifying x = 2.5

Subproblem 2

x + -0.875 = -1.625 Simplifying x + -0.875 = -1.625 Reorder the terms: -0.875 + x = -1.625 Solving -0.875 + x = -1.625 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 = -1.625 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + x = -1.625 + 0.875 x = -1.625 + 0.875 Combine like terms: -1.625 + 0.875 = -0.75 x = -0.75 Simplifying x = -0.75

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.5, -0.75}

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