3x(x-5)=7x^2-16

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Solution for 3x(x-5)=7x^2-16 equation:


Simplifying
3x(x + -5) = 7x2 + -16

Reorder the terms:
3x(-5 + x) = 7x2 + -16
(-5 * 3x + x * 3x) = 7x2 + -16
(-15x + 3x2) = 7x2 + -16

Reorder the terms:
-15x + 3x2 = -16 + 7x2

Solving
-15x + 3x2 = -16 + 7x2

Solving for variable 'x'.

Reorder the terms:
16 + -15x + 3x2 + -7x2 = -16 + 7x2 + 16 + -7x2

Combine like terms: 3x2 + -7x2 = -4x2
16 + -15x + -4x2 = -16 + 7x2 + 16 + -7x2

Reorder the terms:
16 + -15x + -4x2 = -16 + 16 + 7x2 + -7x2

Combine like terms: -16 + 16 = 0
16 + -15x + -4x2 = 0 + 7x2 + -7x2
16 + -15x + -4x2 = 7x2 + -7x2

Combine like terms: 7x2 + -7x2 = 0
16 + -15x + -4x2 = 0

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

Divide each side by '-4'.
-4 + 3.75x + x2 = 0

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + 3.75x + 4 + x2 = 0 + 4

Reorder the terms:
-4 + 4 + 3.75x + x2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + 3.75x + x2 = 0 + 4
3.75x + x2 = 0 + 4

Combine like terms: 0 + 4 = 4
3.75x + x2 = 4

The x term is 3.75x.  Take half its coefficient (1.875).
Square it (3.515625) and add it to both sides.

Add '3.515625' to each side of the equation.
3.75x + 3.515625 + x2 = 4 + 3.515625

Reorder the terms:
3.515625 + 3.75x + x2 = 4 + 3.515625

Combine like terms: 4 + 3.515625 = 7.515625
3.515625 + 3.75x + x2 = 7.515625

Factor a perfect square on the left side:
(x + 1.875)(x + 1.875) = 7.515625

Calculate the square root of the right side: 2.741464025

Break this problem into two subproblems by setting 
(x + 1.875) equal to 2.741464025 and -2.741464025.

Subproblem 1

x + 1.875 = 2.741464025 Simplifying x + 1.875 = 2.741464025 Reorder the terms: 1.875 + x = 2.741464025 Solving 1.875 + x = 2.741464025 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.875' to each side of the equation. 1.875 + -1.875 + x = 2.741464025 + -1.875 Combine like terms: 1.875 + -1.875 = 0.000 0.000 + x = 2.741464025 + -1.875 x = 2.741464025 + -1.875 Combine like terms: 2.741464025 + -1.875 = 0.866464025 x = 0.866464025 Simplifying x = 0.866464025

Subproblem 2

x + 1.875 = -2.741464025 Simplifying x + 1.875 = -2.741464025 Reorder the terms: 1.875 + x = -2.741464025 Solving 1.875 + x = -2.741464025 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.875' to each side of the equation. 1.875 + -1.875 + x = -2.741464025 + -1.875 Combine like terms: 1.875 + -1.875 = 0.000 0.000 + x = -2.741464025 + -1.875 x = -2.741464025 + -1.875 Combine like terms: -2.741464025 + -1.875 = -4.616464025 x = -4.616464025 Simplifying x = -4.616464025

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.866464025, -4.616464025}

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