Algebra
Calculus
Trigonometry
Matrix
Question
Solve the inequality
a∈(4−22,2)∪(2,4+22)
Evaluate
2∣2a−4∣×8a2−4(2a−4)2>0
Find the domain
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Evaluate
8a2−4(2a−4)2≥0
Calculate
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Evaluate
8a2−4(2a−4)2
Expand the expression
8a2−16a2+64a−64
Subtract the terms
−8a2+64a−64
−8a2+64a−64≥0
Move the constant to the right side
−8a2+64a≥0−(−64)
Add the terms
−8a2+64a≥64
Evaluate
a2−8a≤−8
Add the same value to both sides
a2−8a+16≤−8+16
Evaluate
a2−8a+16≤8
Evaluate
(a−4)2≤8
Take the 2-th root on both sides of the inequality
(a−4)2≤8
Calculate
∣a−4∣≤22
Separate the inequality into 2 possible cases
{a−4≤22a−4≥−22
Calculate
{a≤22+4a−4≥−22
Calculate
{a≤22+4a≥−22+4
Find the intersection
−22+4≤a≤22+4
2∣2a−4∣×8a2−4(2a−4)2>0,−22+4≤a≤22+4
Simplify
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Evaluate
2∣2a−4∣×8a2−4(2a−4)2
Subtract the terms
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Simplify
8a2−4(2a−4)2
Expand the expression
8a2−16a2+64a−64
Subtract the terms
−8a2+64a−64
2∣2a−4∣×−8a2+64a−64
Calculate the absolute value
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Calculate
∣2a−4∣
Rewrite the expression
∣2(a−2)∣
Rewrite the expression
2∣a−2∣
2×2∣a−2∣×−8a2+64a−64
Simplify the root
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Evaluate
−8a2+64a−64
Factor the expression
8(−a2+8a−8)
The root of a product is equal to the product of the roots of each factor
8×−a2+8a−8
Evaluate the root
22×−a2+8a−8
Calculate the product
2−2a2+16a−16
2×2∣a−2∣×2−2a2+16a−16
Multiply the terms with the same base by adding their exponents
21+1+1∣a−2∣×−2a2+16a−16
Add the numbers
23∣a−2∣×−2a2+16a−16
Multiply the terms
23−2a2+16a−16×∣a−2∣
23−2a2+16a−16×∣a−2∣>0
Since the left-hand side is always positive or 0,and the right-hand side is always 0,the statement is true for any value of a,except when 23−2a2+16a−16×∣a−2∣=0
23−2a2+16a−16×∣a−2∣=0
Elimination the left coefficient
−2a2+16a−16×∣a−2∣=0
Separate the equation into 2 possible cases
−2a2+16a−16=0∣a−2∣=0
Solve the equation
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Evaluate
−2a2+16a−16=0
The only way a root could be 0 is when the radicand equals 0
−2a2+16a−16=0
Multiply both sides
2a2−16a+16=0
Substitute a=2,b=−16 and c=16 into the quadratic formula a=2a−b±b2−4ac
a=2×216±(−16)2−4×2×16
Simplify the expression
a=416±(−16)2−4×2×16
Simplify the expression
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Evaluate
(−16)2−4×2×16
Multiply the terms
(−16)2−128
Rewrite the expression
162−128
Evaluate the power
256−128
Subtract the numbers
128
a=416±128
Simplify the radical expression
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Evaluate
128
Write the expression as a product where the root of one of the factors can be evaluated
64×2
Write the number in exponential form with the base of 8
82×2
The root of a product is equal to the product of the roots of each factor
82×2
Reduce the index of the radical and exponent with 2
82
a=416±82
Separate the equation into 2 possible cases
a=416+82a=416−82
Simplify the expression
a=4+22a=416−82
Simplify the expression
a=4+22a=4−22
a=4+22a=4−22∣a−2∣=0
Solve the equation
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Evaluate
∣a−2∣=0
Rewrite the expression
a−2=0
Move the constant to the right-hand side and change its sign
a=0+2
Removing 0 doesn't change the value,so remove it from the expression
a=2
a=4+22a=4−22a=2
Exclude the impossible values of a
a∈(−∞,4−22)∪(4−22,2)∪(2,4+22)∪(4+22,+∞)
Check if the solution is in the defined range
a∈(−∞,4−22)∪(4−22,2)∪(2,4+22)∪(4+22,+∞),−22+4≤a≤22+4
Solution
a∈(4−22,2)∪(2,4+22)
Show Solution
