30z^2-35z^2-24z+28=0

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Solution for 30z^2-35z^2-24z+28=0 equation:


Simplifying
30z2 + -35z2 + -24z + 28 = 0

Reorder the terms:
28 + -24z + 30z2 + -35z2 = 0

Combine like terms: 30z2 + -35z2 = -5z2
28 + -24z + -5z2 = 0

Solving
28 + -24z + -5z2 = 0

Solving for variable 'z'.

Begin completing the square.  Divide all terms by
-5 the coefficient of the squared term: 

Divide each side by '-5'.
-5.6 + 4.8z + z2 = 0

Move the constant term to the right:

Add '5.6' to each side of the equation.
-5.6 + 4.8z + 5.6 + z2 = 0 + 5.6

Reorder the terms:
-5.6 + 5.6 + 4.8z + z2 = 0 + 5.6

Combine like terms: -5.6 + 5.6 = 0.0
0.0 + 4.8z + z2 = 0 + 5.6
4.8z + z2 = 0 + 5.6

Combine like terms: 0 + 5.6 = 5.6
4.8z + z2 = 5.6

The z term is 4.8z.  Take half its coefficient (2.4).
Square it (5.76) and add it to both sides.

Add '5.76' to each side of the equation.
4.8z + 5.76 + z2 = 5.6 + 5.76

Reorder the terms:
5.76 + 4.8z + z2 = 5.6 + 5.76

Combine like terms: 5.6 + 5.76 = 11.36
5.76 + 4.8z + z2 = 11.36

Factor a perfect square on the left side:
(z + 2.4)(z + 2.4) = 11.36

Calculate the square root of the right side: 3.370459909

Break this problem into two subproblems by setting 
(z + 2.4) equal to 3.370459909 and -3.370459909.

Subproblem 1

z + 2.4 = 3.370459909 Simplifying z + 2.4 = 3.370459909 Reorder the terms: 2.4 + z = 3.370459909 Solving 2.4 + z = 3.370459909 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2.4' to each side of the equation. 2.4 + -2.4 + z = 3.370459909 + -2.4 Combine like terms: 2.4 + -2.4 = 0.0 0.0 + z = 3.370459909 + -2.4 z = 3.370459909 + -2.4 Combine like terms: 3.370459909 + -2.4 = 0.970459909 z = 0.970459909 Simplifying z = 0.970459909

Subproblem 2

z + 2.4 = -3.370459909 Simplifying z + 2.4 = -3.370459909 Reorder the terms: 2.4 + z = -3.370459909 Solving 2.4 + z = -3.370459909 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2.4' to each side of the equation. 2.4 + -2.4 + z = -3.370459909 + -2.4 Combine like terms: 2.4 + -2.4 = 0.0 0.0 + z = -3.370459909 + -2.4 z = -3.370459909 + -2.4 Combine like terms: -3.370459909 + -2.4 = -5.770459909 z = -5.770459909 Simplifying z = -5.770459909

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.970459909, -5.770459909}

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