a(2a-5)+7(a-3)=

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Solution for a(2a-5)+7(a-3)= equation:


Simplifying
a(2a + -5) + 7(a + -3) = 0

Reorder the terms:
a(-5 + 2a) + 7(a + -3) = 0
(-5 * a + 2a * a) + 7(a + -3) = 0
(-5a + 2a2) + 7(a + -3) = 0

Reorder the terms:
-5a + 2a2 + 7(-3 + a) = 0
-5a + 2a2 + (-3 * 7 + a * 7) = 0
-5a + 2a2 + (-21 + 7a) = 0

Reorder the terms:
-21 + -5a + 7a + 2a2 = 0

Combine like terms: -5a + 7a = 2a
-21 + 2a + 2a2 = 0

Solving
-21 + 2a + 2a2 = 0

Solving for variable 'a'.

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

Divide each side by '2'.
-10.5 + a + a2 = 0

Move the constant term to the right:

Add '10.5' to each side of the equation.
-10.5 + a + 10.5 + a2 = 0 + 10.5

Reorder the terms:
-10.5 + 10.5 + a + a2 = 0 + 10.5

Combine like terms: -10.5 + 10.5 = 0.0
0.0 + a + a2 = 0 + 10.5
a + a2 = 0 + 10.5

Combine like terms: 0 + 10.5 = 10.5
a + a2 = 10.5

The a term is a.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
 + 0.25 + a2 = 10.5 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + a2 = 10.5 + 0.25

Combine like terms: 10.5 + 0.25 = 10.75
1.25 + a2 = 10.75

Factor a perfect square on the left side:
(a + 0.5)(a + 0.5) = 10.75

Calculate the square root of the right side: 3.278719262

Break this problem into two subproblems by setting 
(a + 0.5) equal to 3.278719262 and -3.278719262.

Subproblem 1

a + 0.5 = 3.278719262 Simplifying a + 0.5 = 3.278719262 Reorder the terms: 0.5 + a = 3.278719262 Solving 0.5 + a = 3.278719262 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + a = 3.278719262 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = 3.278719262 + -0.5 a = 3.278719262 + -0.5 Combine like terms: 3.278719262 + -0.5 = 2.778719262 a = 2.778719262 Simplifying a = 2.778719262

Subproblem 2

a + 0.5 = -3.278719262 Simplifying a + 0.5 = -3.278719262 Reorder the terms: 0.5 + a = -3.278719262 Solving 0.5 + a = -3.278719262 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + a = -3.278719262 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = -3.278719262 + -0.5 a = -3.278719262 + -0.5 Combine like terms: -3.278719262 + -0.5 = -3.778719262 a = -3.778719262 Simplifying a = -3.778719262

Solution

The solution to the problem is based on the solutions from the subproblems. a = {2.778719262, -3.778719262}

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