Solve for a 4a^4=8a^3+4a^2

Math
4a4=8a3+4a2
Since a is on the right side of the equation, switch the sides so it is on the left side of the equation.
8a3+4a2=4a4
Subtract 4a4 from both sides of the equation.
8a3+4a2-4a4=0
Factor -4a2 out of 8a3+4a2-4a4.
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Reorder the expression.
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Move 4a2.
8a3-4a4+4a2=0
Reorder 8a3 and -4a4.
-4a4+8a3+4a2=0
-4a4+8a3+4a2=0
Factor -4a2 out of -4a4.
-4a2a2+8a3+4a2=0
Factor -4a2 out of 8a3.
-4a2a2-4a2(-2a)+4a2=0
Factor -4a2 out of 4a2.
-4a2a2-4a2(-2a)-4a2⋅-1=0
Factor -4a2 out of -4a2(a2)-4a2(-2a).
-4a2(a2-2a)-4a2⋅-1=0
Factor -4a2 out of -4a2(a2-2a)-4a2(-1).
-4a2(a2-2a-1)=0
-4a2(a2-2a-1)=0
Divide each term by -4 and simplify.
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Divide each term in -4a2(a2-2a-1)=0 by -4.
-4a2(a2-2a-1)-4=0-4
Simplify -4a2(a2-2a-1)-4.
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Cancel the common factor of -4.
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Cancel the common factor.
-4a2(a2-2a-1)-4=0-4
Divide a2(a2-2a-1) by 1.
a2(a2-2a-1)=0-4
a2(a2-2a-1)=0-4
Apply the distributive property.
a2a2+a2(-2a)+a2⋅-1=0-4
Simplify.
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Multiply a2 by a2 by adding the exponents.
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Use the power rule aman=am+n to combine exponents.
a2+2+a2(-2a)+a2⋅-1=0-4
Add 2 and 2.
a4+a2(-2a)+a2⋅-1=0-4
a4+a2(-2a)+a2⋅-1=0-4
Rewrite using the commutative property of multiplication.
a4-2a2a+a2⋅-1=0-4
Move -1 to the left of a2.
a4-2a2a-1⋅a2=0-4
a4-2a2a-1⋅a2=0-4
Simplify each term.
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Multiply a2 by a by adding the exponents.
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Move a.
a4-2(a⋅a2)-1⋅a2=0-4
Multiply a by a2.
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Raise a to the power of 1.
a4-2(a1a2)-1⋅a2=0-4
Use the power rule aman=am+n to combine exponents.
a4-2a1+2-1⋅a2=0-4
a4-2a1+2-1⋅a2=0-4
Add 1 and 2.
a4-2a3-1⋅a2=0-4
a4-2a3-1⋅a2=0-4
Rewrite -1a2 as -a2.
a4-2a3-a2=0-4
a4-2a3-a2=0-4
a4-2a3-a2=0-4
Divide 0 by -4.
a4-2a3-a2=0
a4-2a3-a2=0
Factor a2 out of a4-2a3-a2.
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Factor a2 out of a4.
a2a2-2a3-a2=0
Factor a2 out of -2a3.
a2a2+a2(-2a)-a2=0
Factor a2 out of -a2.
a2a2+a2(-2a)+a2⋅-1=0
Factor a2 out of a2a2+a2(-2a).
a2(a2-2a)+a2⋅-1=0
Factor a2 out of a2(a2-2a)+a2⋅-1.
a2(a2-2a-1)=0
a2(a2-2a-1)=0
If any individual factor on the left side of the equation is equal to 0, the entire expression will be equal to 0.
a2=0
a2-2a-1=0
Set the first factor equal to 0 and solve.
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Set the first factor equal to 0.
a2=0
Take the square root of both sides of the equation to eliminate the exponent on the left side.
a=±0
The complete solution is the result of both the positive and negative portions of the solution.
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Simplify the right side of the equation.
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Rewrite 0 as 02.
a=±02
Pull terms out from under the radical, assuming positive real numbers.
a=±0
a=±0
±0 is equal to 0.
a=0
a=0
a=0
Set the next factor equal to 0 and solve.
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Set the next factor equal to 0.
a2-2a-1=0
Use the quadratic formula to find the solutions.
-b±b2-4(ac)2a
Substitute the values a=1, b=-2, and c=-1 into the quadratic formula and solve for a.
2±(-2)2-4⋅(1⋅-1)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⋅-1)2⋅1
Multiply -1 by 1.
a=2±4-4⋅-12⋅1
Multiply -4 by -1.
a=2±4+42⋅1
Add 4 and 4.
a=2±82⋅1
Rewrite 8 as 22⋅2.
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Factor 4 out of 8.
a=2±4(2)2⋅1
Rewrite 4 as 22.
a=2±22⋅22⋅1
a=2±22⋅22⋅1
Pull terms out from under the radical.
a=2±222⋅1
a=2±222⋅1
Multiply 2 by 1.
a=2±222
Simplify 2±222.
a=1±2
a=1±2
The final answer is the combination of both solutions.
a=1+2,1-2
a=1+2,1-2
The final solution is all the values that make a2(a2-2a-1)=0 true.
a=0,1+2,1-2
The result can be shown in multiple forms.
Exact Form:
a=0,1+2,1-2
Decimal Form:
a=0,2.41421356…,-0.41421356…
Solve for a 4a^4=8a^3+4a^2

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