49z^2+42z=19

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Solution for 49z^2+42z=19 equation:


Simplifying
49z2 + 42z = 19

Reorder the terms:
42z + 49z2 = 19

Solving
42z + 49z2 = 19

Solving for variable 'z'.

Reorder the terms:
-19 + 42z + 49z2 = 19 + -19

Combine like terms: 19 + -19 = 0
-19 + 42z + 49z2 = 0

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

Divide each side by '49'.
-0.387755102 + 0.8571428571z + z2 = 0

Move the constant term to the right:

Add '0.387755102' to each side of the equation.
-0.387755102 + 0.8571428571z + 0.387755102 + z2 = 0 + 0.387755102

Reorder the terms:
-0.387755102 + 0.387755102 + 0.8571428571z + z2 = 0 + 0.387755102

Combine like terms: -0.387755102 + 0.387755102 = 0.000000000
0.000000000 + 0.8571428571z + z2 = 0 + 0.387755102
0.8571428571z + z2 = 0 + 0.387755102

Combine like terms: 0 + 0.387755102 = 0.387755102
0.8571428571z + z2 = 0.387755102

The z term is 0.8571428571z.  Take half its coefficient (0.4285714286).
Square it (0.1836734694) and add it to both sides.

Add '0.1836734694' to each side of the equation.
0.8571428571z + 0.1836734694 + z2 = 0.387755102 + 0.1836734694

Reorder the terms:
0.1836734694 + 0.8571428571z + z2 = 0.387755102 + 0.1836734694

Combine like terms: 0.387755102 + 0.1836734694 = 0.5714285714
0.1836734694 + 0.8571428571z + z2 = 0.5714285714

Factor a perfect square on the left side:
(z + 0.4285714286)(z + 0.4285714286) = 0.5714285714

Calculate the square root of the right side: 0.755928946

Break this problem into two subproblems by setting 
(z + 0.4285714286) equal to 0.755928946 and -0.755928946.

Subproblem 1

z + 0.4285714286 = 0.755928946 Simplifying z + 0.4285714286 = 0.755928946 Reorder the terms: 0.4285714286 + z = 0.755928946 Solving 0.4285714286 + z = 0.755928946 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.4285714286' to each side of the equation. 0.4285714286 + -0.4285714286 + z = 0.755928946 + -0.4285714286 Combine like terms: 0.4285714286 + -0.4285714286 = 0.0000000000 0.0000000000 + z = 0.755928946 + -0.4285714286 z = 0.755928946 + -0.4285714286 Combine like terms: 0.755928946 + -0.4285714286 = 0.3273575174 z = 0.3273575174 Simplifying z = 0.3273575174

Subproblem 2

z + 0.4285714286 = -0.755928946 Simplifying z + 0.4285714286 = -0.755928946 Reorder the terms: 0.4285714286 + z = -0.755928946 Solving 0.4285714286 + z = -0.755928946 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.4285714286' to each side of the equation. 0.4285714286 + -0.4285714286 + z = -0.755928946 + -0.4285714286 Combine like terms: 0.4285714286 + -0.4285714286 = 0.0000000000 0.0000000000 + z = -0.755928946 + -0.4285714286 z = -0.755928946 + -0.4285714286 Combine like terms: -0.755928946 + -0.4285714286 = -1.1845003746 z = -1.1845003746 Simplifying z = -1.1845003746

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.3273575174, -1.1845003746}

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