10r^2=15-7r

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Solution for 10r^2=15-7r equation:


Simplifying
10r2 = 15 + -7r

Solving
10r2 = 15 + -7r

Solving for variable 'r'.

Reorder the terms:
-15 + 7r + 10r2 = 15 + -7r + -15 + 7r

Reorder the terms:
-15 + 7r + 10r2 = 15 + -15 + -7r + 7r

Combine like terms: 15 + -15 = 0
-15 + 7r + 10r2 = 0 + -7r + 7r
-15 + 7r + 10r2 = -7r + 7r

Combine like terms: -7r + 7r = 0
-15 + 7r + 10r2 = 0

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

Divide each side by '10'.
-1.5 + 0.7r + r2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + 0.7r + 1.5 + r2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + 0.7r + r2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + 0.7r + r2 = 0 + 1.5
0.7r + r2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
0.7r + r2 = 1.5

The r term is 0.7r.  Take half its coefficient (0.35).
Square it (0.1225) and add it to both sides.

Add '0.1225' to each side of the equation.
0.7r + 0.1225 + r2 = 1.5 + 0.1225

Reorder the terms:
0.1225 + 0.7r + r2 = 1.5 + 0.1225

Combine like terms: 1.5 + 0.1225 = 1.6225
0.1225 + 0.7r + r2 = 1.6225

Factor a perfect square on the left side:
(r + 0.35)(r + 0.35) = 1.6225

Calculate the square root of the right side: 1.27377392

Break this problem into two subproblems by setting 
(r + 0.35) equal to 1.27377392 and -1.27377392.

Subproblem 1

r + 0.35 = 1.27377392 Simplifying r + 0.35 = 1.27377392 Reorder the terms: 0.35 + r = 1.27377392 Solving 0.35 + r = 1.27377392 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.35' to each side of the equation. 0.35 + -0.35 + r = 1.27377392 + -0.35 Combine like terms: 0.35 + -0.35 = 0.00 0.00 + r = 1.27377392 + -0.35 r = 1.27377392 + -0.35 Combine like terms: 1.27377392 + -0.35 = 0.92377392 r = 0.92377392 Simplifying r = 0.92377392

Subproblem 2

r + 0.35 = -1.27377392 Simplifying r + 0.35 = -1.27377392 Reorder the terms: 0.35 + r = -1.27377392 Solving 0.35 + r = -1.27377392 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.35' to each side of the equation. 0.35 + -0.35 + r = -1.27377392 + -0.35 Combine like terms: 0.35 + -0.35 = 0.00 0.00 + r = -1.27377392 + -0.35 r = -1.27377392 + -0.35 Combine like terms: -1.27377392 + -0.35 = -1.62377392 r = -1.62377392 Simplifying r = -1.62377392

Solution

The solution to the problem is based on the solutions from the subproblems. r = {0.92377392, -1.62377392}

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