Question
Simplify the expression
∣x−2∣x2−4x+3−2∣x−2∣
Evaluate
∣x−2∣−∣x−2∣1−2
Subtract the terms
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Evaluate
∣x−2∣−∣x−2∣1
Reduce fractions to a common denominator
∣x−2∣∣x−2∣∣x−2∣−∣x−2∣1
Write all numerators above the common denominator
∣x−2∣∣x−2∣∣x−2∣−1
Multiply the terms
∣x−2∣(x−2)2−1
Expand the expression
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Evaluate
(x−2)2
Use (a−b)2=a2−2ab+b2 to expand the expression
x2−2x×2+22
Calculate
x2−4x+4
∣x−2∣x2−4x+4−1
Subtract the numbers
∣x−2∣x2−4x+3
∣x−2∣x2−4x+3−2
Reduce fractions to a common denominator
∣x−2∣x2−4x+3−∣x−2∣2∣x−2∣
Solution
∣x−2∣x2−4x+3−2∣x−2∣
Show Solution

Find the excluded values
x=2
Evaluate
∣x−2∣−∣x−2∣1−2
To find the excluded values,set the denominators equal to 0
∣x−2∣=0
Rewrite the expression
x−2=0
Move the constant to the right-hand side and change its sign
x=0+2
Solution
x=2
Show Solution

Find the roots
x1=1−2,x2=3+2
Alternative Form
x1≈−0.414214,x2≈4.414214
Evaluate
∣x−2∣−∣x−2∣1−2
To find the roots of the expression,set the expression equal to 0
∣x−2∣−∣x−2∣1−2=0
Find the domain
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Evaluate
∣x−2∣=0
Rewrite the expression
x−2=0
Move the constant to the right side
x=0+2
Removing 0 doesn't change the value,so remove it from the expression
x=2
∣x−2∣−∣x−2∣1−2=0,x=2
Calculate
∣x−2∣−∣x−2∣1−2=0
Subtract the terms
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Simplify
∣x−2∣−∣x−2∣1
Reduce fractions to a common denominator
∣x−2∣∣x−2∣∣x−2∣−∣x−2∣1
Write all numerators above the common denominator
∣x−2∣∣x−2∣∣x−2∣−1
Multiply the terms
∣x−2∣(x−2)2−1
Expand the expression
More Steps

Evaluate
(x−2)2
Use (a−b)2=a2−2ab+b2 to expand the expression
x2−2x×2+22
Calculate
x2−4x+4
∣x−2∣x2−4x+4−1
Subtract the numbers
∣x−2∣x2−4x+3
∣x−2∣x2−4x+3−2=0
Subtract the terms
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Simplify
∣x−2∣x2−4x+3−2
Reduce fractions to a common denominator
∣x−2∣x2−4x+3−∣x−2∣2∣x−2∣
Write all numerators above the common denominator
∣x−2∣x2−4x+3−2∣x−2∣
∣x−2∣x2−4x+3−2∣x−2∣=0
Cross multiply
x2−4x+3−2∣x−2∣=∣x−2∣×0
Simplify the equation
x2−4x+3−2∣x−2∣=0
Separate the equation into 2 possible cases
x2−4x+3−2(x−2)=0,x−2≥0x2−4x+3−2(−(x−2))=0,x−2<0
Solve the equation
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Evaluate
x2−4x+3−2(x−2)=0
Calculate the sum or difference
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Evaluate
x2−4x+3−2(x−2)
Expand the expression
x2−4x+3−2x+4
Subtract the terms
x2−6x+3+4
Add the numbers
x2−6x+7
x2−6x+7=0
Substitute a=1,b=−6 and c=7 into the quadratic formula x=2a−b±b2−4ac
x=26±(−6)2−4×7
Simplify the expression
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Evaluate
(−6)2−4×7
Multiply the numbers
(−6)2−28
Rewrite the expression
62−28
Evaluate the power
36−28
Subtract the numbers
8
x=26±8
Simplify the radical expression
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Evaluate
8
Write the expression as a product where the root of one of the factors can be evaluated
4×2
Write the number in exponential form with the base of 2
22×2
The root of a product is equal to the product of the roots of each factor
22×2
Reduce the index of the radical and exponent with 2
22
x=26±22
Separate the equation into 2 possible cases
x=26+22x=26−22
Simplify the expression
x=3+2x=26−22
Simplify the expression
x=3+2x=3−2
x=3+2x=3−2,x−2≥0x2−4x+3−2(−(x−2))=0,x−2<0
Solve the inequality
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Evaluate
x−2≥0
Move the constant to the right side
x≥0+2
Removing 0 doesn't change the value,so remove it from the expression
x≥2
x=3+2x=3−2,x≥2x2−4x+3−2(−(x−2))=0,x−2<0
Solve the equation
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Evaluate
x2−4x+3−2(−(x−2))=0
Calculate
x2−4x+3−2(−x+2)=0
Calculate the sum or difference
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Evaluate
x2−4x+3−2(−x+2)
Expand the expression
x2−4x+3+2x−4
Add the terms
x2−2x+3−4
Subtract the numbers
x2−2x−1
x2−2x−1=0
Substitute a=1,b=−2 and c=−1 into the quadratic formula x=2a−b±b2−4ac
x=22±(−2)2−4(−1)
Simplify the expression
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Evaluate
(−2)2−4(−1)
Simplify
(−2)2−(−4)
Rewrite the expression
22−(−4)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
22+4
Evaluate the power
4+4
Add the numbers
8
x=22±8
Simplify the radical expression
More Steps

Evaluate
8
Write the expression as a product where the root of one of the factors can be evaluated
4×2
Write the number in exponential form with the base of 2
22×2
The root of a product is equal to the product of the roots of each factor
22×2
Reduce the index of the radical and exponent with 2
22
x=22±22
Separate the equation into 2 possible cases
x=22+22x=22−22
Simplify the expression
x=1+2x=22−22
Simplify the expression
x=1+2x=1−2
x=3+2x=3−2,x≥2x=1+2x=1−2,x−2<0
Solve the inequality
More Steps

Evaluate
x−2<0
Move the constant to the right side
x<0+2
Removing 0 doesn't change the value,so remove it from the expression
x<2
x=3+2x=3−2,x≥2x=1+2x=1−2,x<2
Find the intersection
x=3+2x=1+2x=1−2,x<2
Find the intersection
x=3+2x=1−2
Check if the solution is in the defined range
x=3+2x=1−2,x=2
Find the intersection of the solution and the defined range
x=3+2x=1−2
Solution
x1=1−2,x2=3+2
Alternative Form
x1≈−0.414214,x2≈4.414214
Show Solution
