Solve for a |a^2-3a+6|=|a-3|

Math
|a2-3a+6|=|a-3|
Rewrite the absolute value equation as four equations without absolute value bars.
a2-3a+6=a-3
a2-3a+6=-(a-3)
-(a2-3a+6)=a-3
-(a2-3a+6)=-(a-3)
After simplifying, there are only two unique equations to be solved.
a2-3a+6=a-3
a2-3a+6=-(a-3)
Solve a2-3a+6=a-3 for a.
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Move all terms containing a to the left side of the equation.
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Subtract a from both sides of the equation.
a2-3a+6-a=-3
Subtract a from -3a.
a2-4a+6=-3
a2-4a+6=-3
Move all terms to the left side of the equation and simplify.
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Move 3 to the left side of the equation by adding it to both sides.
a2-4a+6+3=0
Add 6 and 3.
a2-4a+9=0
a2-4a+9=0
Use the quadratic formula to find the solutions.
-b±b2-4(ac)2a
Substitute the values a=1, b=-4, and c=9 into the quadratic formula and solve for a.
4±(-4)2-4⋅(1⋅9)2⋅1
Simplify.
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Simplify the numerator.
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Raise -4 to the power of 2.
a=4±16-4⋅(1⋅9)2⋅1
Multiply 9 by 1.
a=4±16-4⋅92⋅1
Multiply -4 by 9.
a=4±16-362⋅1
Subtract 36 from 16.
a=4±-202⋅1
Rewrite -20 as -1(20).
a=4±-1⋅202⋅1
Rewrite -1(20) as -1⋅20.
a=4±-1⋅202⋅1
Rewrite -1 as i.
a=4±i⋅202⋅1
Rewrite 20 as 22⋅5.
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Factor 4 out of 20.
a=4±i⋅4(5)2⋅1
Rewrite 4 as 22.
a=4±i⋅22⋅52⋅1
a=4±i⋅22⋅52⋅1
Pull terms out from under the radical.
a=4±i⋅(25)2⋅1
Move 2 to the left of i.
a=4±2i52⋅1
a=4±2i52⋅1
Multiply 2 by 1.
a=4±2i52
Simplify 4±2i52.
a=2±i5
a=2±i5
The final answer is the combination of both solutions.
a=2+i5,2-i5
a=2+i5,2-i5
Solve a2-3a+6=-(a-3) for a.
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Simplify -(a-3).
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Apply the distributive property.
a2-3a+6=-a–3
Multiply -1 by -3.
a2-3a+6=-a+3
a2-3a+6=-a+3
Move all terms containing a to the left side of the equation.
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Add a to both sides of the equation.
a2-3a+6+a=3
Add -3a and a.
a2-2a+6=3
a2-2a+6=3
Move all terms to the left side of the equation and simplify.
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Move 3 to the left side of the equation by subtracting it from both sides.
a2-2a+6-3=0
Subtract 3 from 6.
a2-2a+3=0
a2-2a+3=0
Use the quadratic formula to find the solutions.
-b±b2-4(ac)2a
Substitute the values a=1, b=-2, and c=3 into the quadratic formula and solve for a.
2±(-2)2-4⋅(1⋅3)2⋅1
Simplify.
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Simplify the numerator.
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Raise -2 to the power of 2.
a=2±4-4⋅(1⋅3)2⋅1
Multiply 3 by 1.
a=2±4-4⋅32⋅1
Multiply -4 by 3.
a=2±4-122⋅1
Subtract 12 from 4.
a=2±-82⋅1
Rewrite -8 as -1(8).
a=2±-1⋅82⋅1
Rewrite -1(8) as -1⋅8.
a=2±-1⋅82⋅1
Rewrite -1 as i.
a=2±i⋅82⋅1
Rewrite 8 as 22⋅2.
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Factor 4 out of 8.
a=2±i⋅4(2)2⋅1
Rewrite 4 as 22.
a=2±i⋅22⋅22⋅1
a=2±i⋅22⋅22⋅1
Pull terms out from under the radical.
a=2±i⋅(22)2⋅1
Move 2 to the left of i.
a=2±2i22⋅1
a=2±2i22⋅1
Multiply 2 by 1.
a=2±2i22
Simplify 2±2i22.
a=1±i2
a=1±i2
The final answer is the combination of both solutions.
a=1+i2,1-i2
a=1+i2,1-i2
List all of the solutions.
a=
Solve for a |a^2-3a+6|=|a-3|

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