# Solve for a a*a+b*b=6ab

a⋅a+b⋅b=6ab
Simplify each term.
Multiply a by a.
a2+b⋅b=6ab
Multiply b by b.
a2+b2=6ab
a2+b2=6ab
Subtract 6ab from both sides of the equation.
a2+b2-6ab=0
Use the quadratic formula to find the solutions.
-b±b2-4(ac)2a
Substitute the values a=1, b=-6b, and c=b2 into the quadratic formula and solve for a.
6b±(-6b)2-4⋅(1⋅b2)2⋅1
Simplify.
Simplify the numerator.
Rewrite 4⋅(1⋅b2) as (2⋅(1b))2.
a=6b±(-6b)2-(2⋅(1b))22⋅1
Since both terms are perfect squares, factor using the difference of squares formula, a2-b2=(a+b)(a-b) where a=-6b and b=2⋅(1b).
a=6b±(-6b+2⋅(1b))(-6b-(2⋅(1b)))2⋅1
Simplify.
Factor 2b out of -6b+2⋅(1b).
Factor 2b out of -6b.
a=6b±(2b(-3)+2⋅(1b))(-6b-(2⋅(1b)))2⋅1
Factor 2b out of 2⋅(1b).
a=6b±(2b(-3)+2b(1))(-6b-(2⋅(1b)))2⋅1
Factor 2b out of 2b(-3)+2b(1).
a=6b±2b(-3+1)(-6b-(2⋅(1b)))2⋅1
a=6b±2b(-3+1)(-6b-(2⋅(1b)))2⋅1
a=6b±2b⋅(-2(-6b-(2⋅(1b))))2⋅1
Multiply -2 by 2.
a=6b±-4b(-6b-(2⋅(1b)))2⋅1
Combine exponents.
Multiply b by 1.
a=6b±-4b(-6b-(2⋅b))2⋅1
Multiply 2 by -1.
a=6b±-4b(-6b-2b)2⋅1
a=6b±-4b(-6b-2b)2⋅1
a=6b±-4b(-6b-2b)2⋅1
Subtract 2b from -6b.
a=6b±-4b(-8b)2⋅1
Multiply -8 by -4.
a=6b±32b⋅b2⋅1
Raise b to the power of 1.
a=6b±32(b⋅b)2⋅1
Raise b to the power of 1.
a=6b±32(b⋅b)2⋅1
Use the power rule aman=am+n to combine exponents.
a=6b±32b1+12⋅1
a=6b±32b22⋅1
Rewrite 32b2 as (4b)2⋅2.
Factor 16 out of 32.
a=6b±16(2)b22⋅1
Rewrite 16 as 42.
a=6b±42⋅(2b2)2⋅1
Move 2.
a=6b±42b2⋅22⋅1
Rewrite 42b2 as (4b)2.
a=6b±(4b)2⋅22⋅1
a=6b±(4b)2⋅22⋅1
Pull terms out from under the radical.
a=6b±4b22⋅1
a=6b±4b22⋅1
Multiply 2 by 1.
a=6b±4b22
Simplify 6b±4b22.
a=3b±2b2
a=3b±2b2
The final answer is the combination of both solutions.
a=3b+2b2
a=3b-2b2
Solve for a a*a+b*b=6ab

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