7a^2+6a-19=2

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Solution for 7a^2+6a-19=2 equation:


Simplifying
7a2 + 6a + -19 = 2

Reorder the terms:
-19 + 6a + 7a2 = 2

Solving
-19 + 6a + 7a2 = 2

Solving for variable 'a'.

Reorder the terms:
-19 + -2 + 6a + 7a2 = 2 + -2

Combine like terms: -19 + -2 = -21
-21 + 6a + 7a2 = 2 + -2

Combine like terms: 2 + -2 = 0
-21 + 6a + 7a2 = 0

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

Divide each side by '7'.
-3 + 0.8571428571a + a2 = 0

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + 0.8571428571a + 3 + a2 = 0 + 3

Reorder the terms:
-3 + 3 + 0.8571428571a + a2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + 0.8571428571a + a2 = 0 + 3
0.8571428571a + a2 = 0 + 3

Combine like terms: 0 + 3 = 3
0.8571428571a + a2 = 3

The a term is 0.8571428571a.  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.
0.8571428571a + 0.1836734694 + a2 = 3 + 0.1836734694

Reorder the terms:
0.1836734694 + 0.8571428571a + a2 = 3 + 0.1836734694

Combine like terms: 3 + 0.1836734694 = 3.1836734694
0.1836734694 + 0.8571428571a + a2 = 3.1836734694

Factor a perfect square on the left side:
(a + 0.4285714286)(a + 0.4285714286) = 3.1836734694

Calculate the square root of the right side: 1.784285142

Break this problem into two subproblems by setting 
(a + 0.4285714286) equal to 1.784285142 and -1.784285142.

Subproblem 1

a + 0.4285714286 = 1.784285142 Simplifying a + 0.4285714286 = 1.784285142 Reorder the terms: 0.4285714286 + a = 1.784285142 Solving 0.4285714286 + a = 1.784285142 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.4285714286' to each side of the equation. 0.4285714286 + -0.4285714286 + a = 1.784285142 + -0.4285714286 Combine like terms: 0.4285714286 + -0.4285714286 = 0.0000000000 0.0000000000 + a = 1.784285142 + -0.4285714286 a = 1.784285142 + -0.4285714286 Combine like terms: 1.784285142 + -0.4285714286 = 1.3557137134 a = 1.3557137134 Simplifying a = 1.3557137134

Subproblem 2

a + 0.4285714286 = -1.784285142 Simplifying a + 0.4285714286 = -1.784285142 Reorder the terms: 0.4285714286 + a = -1.784285142 Solving 0.4285714286 + a = -1.784285142 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.4285714286' to each side of the equation. 0.4285714286 + -0.4285714286 + a = -1.784285142 + -0.4285714286 Combine like terms: 0.4285714286 + -0.4285714286 = 0.0000000000 0.0000000000 + a = -1.784285142 + -0.4285714286 a = -1.784285142 + -0.4285714286 Combine like terms: -1.784285142 + -0.4285714286 = -2.2128565706 a = -2.2128565706 Simplifying a = -2.2128565706

Solution

The solution to the problem is based on the solutions from the subproblems. a = {1.3557137134, -2.2128565706}

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