|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)

Move all terms containing a to the left side of the equation.

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.

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.

Simplify the numerator.

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.

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

Simplify -(a-3).

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.

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.

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.

Simplify the numerator.

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.

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|