r^2+19rs+90s^2=(r+10)(r+9s^2)

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Solution for r^2+19rs+90s^2=(r+10)(r+9s^2) equation:


Simplifying
r2 + 19rs + 90s2 = (r + 10)(r + 9s2)

Reorder the terms:
19rs + r2 + 90s2 = (r + 10)(r + 9s2)

Reorder the terms:
19rs + r2 + 90s2 = (10 + r)(r + 9s2)

Multiply (10 + r) * (r + 9s2)
19rs + r2 + 90s2 = (10(r + 9s2) + r(r + 9s2))
19rs + r2 + 90s2 = ((r * 10 + 9s2 * 10) + r(r + 9s2))
19rs + r2 + 90s2 = ((10r + 90s2) + r(r + 9s2))
19rs + r2 + 90s2 = (10r + 90s2 + (r * r + 9s2 * r))

Reorder the terms:
19rs + r2 + 90s2 = (10r + 90s2 + (9rs2 + r2))
19rs + r2 + 90s2 = (10r + 90s2 + (9rs2 + r2))

Reorder the terms:
19rs + r2 + 90s2 = (10r + 9rs2 + r2 + 90s2)
19rs + r2 + 90s2 = (10r + 9rs2 + r2 + 90s2)

Add '-1r2' to each side of the equation.
19rs + r2 + -1r2 + 90s2 = 10r + 9rs2 + r2 + -1r2 + 90s2

Combine like terms: r2 + -1r2 = 0
19rs + 0 + 90s2 = 10r + 9rs2 + r2 + -1r2 + 90s2
19rs + 90s2 = 10r + 9rs2 + r2 + -1r2 + 90s2

Combine like terms: r2 + -1r2 = 0
19rs + 90s2 = 10r + 9rs2 + 0 + 90s2
19rs + 90s2 = 10r + 9rs2 + 90s2

Add '-90s2' to each side of the equation.
19rs + 90s2 + -90s2 = 10r + 9rs2 + 90s2 + -90s2

Combine like terms: 90s2 + -90s2 = 0
19rs + 0 = 10r + 9rs2 + 90s2 + -90s2
19rs = 10r + 9rs2 + 90s2 + -90s2

Combine like terms: 90s2 + -90s2 = 0
19rs = 10r + 9rs2 + 0
19rs = 10r + 9rs2

Solving
19rs = 10r + 9rs2

Solving for variable 'r'.

Move all terms containing r to the left, all other terms to the right.

Add '-10r' to each side of the equation.
-10r + 19rs = 10r + -10r + 9rs2

Combine like terms: 10r + -10r = 0
-10r + 19rs = 0 + 9rs2
-10r + 19rs = 9rs2

Add '-9rs2' to each side of the equation.
-10r + 19rs + -9rs2 = 9rs2 + -9rs2

Combine like terms: 9rs2 + -9rs2 = 0
-10r + 19rs + -9rs2 = 0

Factor out the Greatest Common Factor (GCF), 'r'.
r(-10 + 19s + -9s2) = 0

Factor a trinomial.
r((-10 + 9s)(1 + -1s)) = 0

Subproblem 1

Set the factor 'r' equal to zero and attempt to solve: Simplifying r = 0 Solving r = 0 Move all terms containing r to the left, all other terms to the right. Simplifying r = 0

Subproblem 2

Set the factor '(-10 + 9s)' equal to zero and attempt to solve: Simplifying -10 + 9s = 0 Solving -10 + 9s = 0 Move all terms containing r to the left, all other terms to the right. Add '10' to each side of the equation. -10 + 10 + 9s = 0 + 10 Combine like terms: -10 + 10 = 0 0 + 9s = 0 + 10 9s = 0 + 10 Combine like terms: 0 + 10 = 10 9s = 10 Add '-9s' to each side of the equation. 9s + -9s = 10 + -9s Combine like terms: 9s + -9s = 0 0 = 10 + -9s Simplifying 0 = 10 + -9s The solution to this equation could not be determined. This subproblem is being ignored because a solution could not be determined.

Subproblem 3

Set the factor '(1 + -1s)' equal to zero and attempt to solve: Simplifying 1 + -1s = 0 Solving 1 + -1s = 0 Move all terms containing r to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + -1s = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -1s = 0 + -1 -1s = 0 + -1 Combine like terms: 0 + -1 = -1 -1s = -1 Add 's' to each side of the equation. -1s + s = -1 + s Combine like terms: -1s + s = 0 0 = -1 + s Simplifying 0 = -1 + s The solution to this equation could not be determined. This subproblem is being ignored because a solution could not be determined.

Solution

r = {0}

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