10z^2-19z-48=0

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Solution for 10z^2-19z-48=0 equation:


Simplifying
10z2 + -19z + -48 = 0

Reorder the terms:
-48 + -19z + 10z2 = 0

Solving
-48 + -19z + 10z2 = 0

Solving for variable 'z'.

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

Divide each side by '10'.
-4.8 + -1.9z + z2 = 0

Move the constant term to the right:

Add '4.8' to each side of the equation.
-4.8 + -1.9z + 4.8 + z2 = 0 + 4.8

Reorder the terms:
-4.8 + 4.8 + -1.9z + z2 = 0 + 4.8

Combine like terms: -4.8 + 4.8 = 0.0
0.0 + -1.9z + z2 = 0 + 4.8
-1.9z + z2 = 0 + 4.8

Combine like terms: 0 + 4.8 = 4.8
-1.9z + z2 = 4.8

The z term is -1.9z.  Take half its coefficient (-0.95).
Square it (0.9025) and add it to both sides.

Add '0.9025' to each side of the equation.
-1.9z + 0.9025 + z2 = 4.8 + 0.9025

Reorder the terms:
0.9025 + -1.9z + z2 = 4.8 + 0.9025

Combine like terms: 4.8 + 0.9025 = 5.7025
0.9025 + -1.9z + z2 = 5.7025

Factor a perfect square on the left side:
(z + -0.95)(z + -0.95) = 5.7025

Calculate the square root of the right side: 2.387990787

Break this problem into two subproblems by setting 
(z + -0.95) equal to 2.387990787 and -2.387990787.

Subproblem 1

z + -0.95 = 2.387990787 Simplifying z + -0.95 = 2.387990787 Reorder the terms: -0.95 + z = 2.387990787 Solving -0.95 + z = 2.387990787 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.95' to each side of the equation. -0.95 + 0.95 + z = 2.387990787 + 0.95 Combine like terms: -0.95 + 0.95 = 0.00 0.00 + z = 2.387990787 + 0.95 z = 2.387990787 + 0.95 Combine like terms: 2.387990787 + 0.95 = 3.337990787 z = 3.337990787 Simplifying z = 3.337990787

Subproblem 2

z + -0.95 = -2.387990787 Simplifying z + -0.95 = -2.387990787 Reorder the terms: -0.95 + z = -2.387990787 Solving -0.95 + z = -2.387990787 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.95' to each side of the equation. -0.95 + 0.95 + z = -2.387990787 + 0.95 Combine like terms: -0.95 + 0.95 = 0.00 0.00 + z = -2.387990787 + 0.95 z = -2.387990787 + 0.95 Combine like terms: -2.387990787 + 0.95 = -1.437990787 z = -1.437990787 Simplifying z = -1.437990787

Solution

The solution to the problem is based on the solutions from the subproblems. z = {3.337990787, -1.437990787}

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