7x^2-61x+18=0

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Solution for 7x^2-61x+18=0 equation:


Simplifying
7x2 + -61x + 18 = 0

Reorder the terms:
18 + -61x + 7x2 = 0

Solving
18 + -61x + 7x2 = 0

Solving for variable 'x'.

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

Divide each side by '7'.
2.571428571 + -8.714285714x + x2 = 0

Move the constant term to the right:

Add '-2.571428571' to each side of the equation.
2.571428571 + -8.714285714x + -2.571428571 + x2 = 0 + -2.571428571

Reorder the terms:
2.571428571 + -2.571428571 + -8.714285714x + x2 = 0 + -2.571428571

Combine like terms: 2.571428571 + -2.571428571 = 0.000000000
0.000000000 + -8.714285714x + x2 = 0 + -2.571428571
-8.714285714x + x2 = 0 + -2.571428571

Combine like terms: 0 + -2.571428571 = -2.571428571
-8.714285714x + x2 = -2.571428571

The x term is -8.714285714x.  Take half its coefficient (-4.357142857).
Square it (18.98469388) and add it to both sides.

Add '18.98469388' to each side of the equation.
-8.714285714x + 18.98469388 + x2 = -2.571428571 + 18.98469388

Reorder the terms:
18.98469388 + -8.714285714x + x2 = -2.571428571 + 18.98469388

Combine like terms: -2.571428571 + 18.98469388 = 16.413265309
18.98469388 + -8.714285714x + x2 = 16.413265309

Factor a perfect square on the left side:
(x + -4.357142857)(x + -4.357142857) = 16.413265309

Calculate the square root of the right side: 4.051328832

Break this problem into two subproblems by setting 
(x + -4.357142857) equal to 4.051328832 and -4.051328832.

Subproblem 1

x + -4.357142857 = 4.051328832 Simplifying x + -4.357142857 = 4.051328832 Reorder the terms: -4.357142857 + x = 4.051328832 Solving -4.357142857 + x = 4.051328832 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4.357142857' to each side of the equation. -4.357142857 + 4.357142857 + x = 4.051328832 + 4.357142857 Combine like terms: -4.357142857 + 4.357142857 = 0.000000000 0.000000000 + x = 4.051328832 + 4.357142857 x = 4.051328832 + 4.357142857 Combine like terms: 4.051328832 + 4.357142857 = 8.408471689 x = 8.408471689 Simplifying x = 8.408471689

Subproblem 2

x + -4.357142857 = -4.051328832 Simplifying x + -4.357142857 = -4.051328832 Reorder the terms: -4.357142857 + x = -4.051328832 Solving -4.357142857 + x = -4.051328832 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4.357142857' to each side of the equation. -4.357142857 + 4.357142857 + x = -4.051328832 + 4.357142857 Combine like terms: -4.357142857 + 4.357142857 = 0.000000000 0.000000000 + x = -4.051328832 + 4.357142857 x = -4.051328832 + 4.357142857 Combine like terms: -4.051328832 + 4.357142857 = 0.305814025 x = 0.305814025 Simplifying x = 0.305814025

Solution

The solution to the problem is based on the solutions from the subproblems. x = {8.408471689, 0.305814025}

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