4x+24x+x^2-5x^2=1

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Solution for 4x+24x+x^2-5x^2=1 equation:


Simplifying
4x + 24x + x2 + -5x2 = 1

Combine like terms: 4x + 24x = 28x
28x + x2 + -5x2 = 1

Combine like terms: x2 + -5x2 = -4x2
28x + -4x2 = 1

Solving
28x + -4x2 = 1

Solving for variable 'x'.

Reorder the terms:
-1 + 28x + -4x2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + 28x + -4x2 = 0

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

Divide each side by '-4'.
0.25 + -7x + x2 = 0

Move the constant term to the right:

Add '-0.25' to each side of the equation.
0.25 + -7x + -0.25 + x2 = 0 + -0.25

Reorder the terms:
0.25 + -0.25 + -7x + x2 = 0 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + -7x + x2 = 0 + -0.25
-7x + x2 = 0 + -0.25

Combine like terms: 0 + -0.25 = -0.25
-7x + x2 = -0.25

The x term is -7x.  Take half its coefficient (-3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
-7x + 12.25 + x2 = -0.25 + 12.25

Reorder the terms:
12.25 + -7x + x2 = -0.25 + 12.25

Combine like terms: -0.25 + 12.25 = 12
12.25 + -7x + x2 = 12

Factor a perfect square on the left side:
(x + -3.5)(x + -3.5) = 12

Calculate the square root of the right side: 3.464101615

Break this problem into two subproblems by setting 
(x + -3.5) equal to 3.464101615 and -3.464101615.

Subproblem 1

x + -3.5 = 3.464101615 Simplifying x + -3.5 = 3.464101615 Reorder the terms: -3.5 + x = 3.464101615 Solving -3.5 + x = 3.464101615 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + x = 3.464101615 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x = 3.464101615 + 3.5 x = 3.464101615 + 3.5 Combine like terms: 3.464101615 + 3.5 = 6.964101615 x = 6.964101615 Simplifying x = 6.964101615

Subproblem 2

x + -3.5 = -3.464101615 Simplifying x + -3.5 = -3.464101615 Reorder the terms: -3.5 + x = -3.464101615 Solving -3.5 + x = -3.464101615 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + x = -3.464101615 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x = -3.464101615 + 3.5 x = -3.464101615 + 3.5 Combine like terms: -3.464101615 + 3.5 = 0.035898385 x = 0.035898385 Simplifying x = 0.035898385

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6.964101615, 0.035898385}

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