Rearrange:
Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :
(a)/(a^2-16)+(2/(a-4))-(2/(a+4))=0
Simplify —————
a + 4
<span>Equation at the end of step 1 :</span><span> a 2 2
(—————————+—————)-——— = 0
((a2)-16) (a-4) a+4
</span><span>Step 2 :</span> 2
Simplify —————
a - 4
<span>Equation at the end of step 2 :</span><span> a 2 2
(—————————+———)-——— = 0
((a2)-16) a-4 a+4
</span><span>Step 3 :</span><span> a
Simplify ———————
a2 - 16
</span>Trying to factor as a Difference of Squares :
<span> 3.1 </span> Factoring: <span> a2 - 16</span>
Theory : A difference of two perfect squares, <span> A2 - B2 </span>can be factored into <span> (A+B) • (A-B)
</span>Proof :<span> (A+B) • (A-B) =
A2 - AB + BA - B2 =
A2 <span>- AB + AB </span>- B2 =
<span> A2 - B2</span>
</span>Note : <span> <span>AB = BA </span></span>is the commutative property of multiplication.
Note : <span> <span>- AB + AB </span></span>equals zero and is therefore eliminated from the expression.
Check : 16 is the square of 4
Check : <span> a2 </span>is the square of <span> a1 </span>
Factorization is : (a + 4) • (a - 4)
<span>Equation at the end of step 3 :</span> a 2 2
(————————————————— + —————) - ————— = 0
(a + 4) • (a - 4) a - 4 a + 4
<span>Step 4 :</span>Calculating the Least Common Multiple :
<span> 4.1 </span> Find the Least Common Multiple
The left denominator is : <span> (a+4) •</span> (a-4)
The right denominator is : <span> a-4 </span>
<span><span> Number of times each Algebraic Factor
appears in the factorization of:</span><span><span><span> Algebraic
Factor </span><span> Left
Denominator </span><span> Right
Denominator </span><span> L.C.M = Max
{Left,Right} </span></span><span><span> a+4 </span>101</span><span><span> a-4 </span>111</span></span></span>
Least Common Multiple:
(a+4) • (a-4)
Calculating Multipliers :
<span> 4.2 </span> Calculate multipliers for the two fractions
Denote the Least Common Multiple by L.C.M
Denote the Left Multiplier by Left_M
Denote the Right Multiplier by Right_M
Denote the Left Deniminator by L_Deno
Denote the Right Multiplier by R_Deno
Left_M = L.C.M / L_Deno = 1
Right_M = L.C.M / R_Deno = a+4
Making Equivalent Fractions :
<span> 4.3 </span> Rewrite the two fractions into<span> equivalent fractions</span>
Two fractions are called <span>equivalent </span>if they have the<span> same numeric value.</span>
For example : 1/2 and 2/4 are equivalent, <span> y/(y+1)2 </span> and <span> (y2+y)/(y+1)3 </span>are equivalent as well.
To calculate equivalent fraction , multiply the <span>Numerator </span>of each fraction, by its respective Multiplier.
<span> L. Mult. • L. Num. a
—————————————————— = —————————————
L.C.M (a+4) • (a-4)
R. Mult. • R. Num. 2 • (a+4)
—————————————————— = —————————————
L.C.M (a+4) • (a-4)
</span>Adding fractions that have a common denominator :
<span> 4.4 </span> Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator
Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:
a + 2 • (a+4) 3a + 8
————————————— = —————————————————
(a+4) • (a-4) (a + 4) • (a - 4)
<span>Equation at the end of step 4 :</span> (3a + 8) 2
————————————————— - ————— = 0
(a + 4) • (a - 4) a + 4
<span>Step 5 :</span>Calculating the Least Common Multiple :
<span> 5.1 </span> Find the Least Common Multiple
The left denominator is : <span> (a+4) •</span> (a-4)
The right denominator is : <span> a+4 </span>
<span><span> Number of times each Algebraic Factor
appears in the factorization of:</span><span><span><span> Algebraic
Factor </span><span> Left
Denominator </span><span> Right
Denominator </span><span> L.C.M = Max
{Left,Right} </span></span><span><span> a+4 </span>111</span><span><span> a-4 </span>101</span></span></span>
Least Common Multiple:
(a+4) • (a-4)
Calculating Multipliers :
<span> 5.2 </span> Calculate multipliers for the two fractions
Denote the Least Common Multiple by L.C.M
Denote the Left Multiplier by Left_M
Denote the Right Multiplier by Right_M
Denote the Left Deniminator by L_Deno
Denote the Right Multiplier by R_Deno
Left_M = L.C.M / L_Deno = 1
Right_M = L.C.M / R_Deno = a-4
Making Equivalent Fractions :
<span> 5.3 </span> Rewrite the two fractions into<span> equivalent fractions</span>
<span> L. Mult. • L. Num. (3a+8)
—————————————————— = —————————————
L.C.M (a+4) • (a-4)
R. Mult. • R. Num. 2 • (a-4)
—————————————————— = —————————————
L.C.M (a+4) • (a-4)
</span>Adding fractions that have a common denominator :
<span> 5.4 </span> Adding up the two equivalent fractions
(3a+8) - (2 • (a-4)) a + 16
———————————————————— = —————————————————
(a+4) • (a-4) (a + 4) • (a - 4)
<span>Equation at the end of step 5 :</span> a + 16
————————————————— = 0
(a + 4) • (a - 4)
<span>Step 6 :</span>When a fraction equals zero :<span><span> 6.1 </span> When a fraction equals zero ...</span>
Where a fraction equals zero, its numerator, the part which is above the fraction line, must equal zero.
Now,to get rid of the <span>denominator, </span>Tiger multiplys both sides of the equation by the denominator.
Here's how:
a+16
——————————— • (a+4)•(a-4) = 0 • (a+4)•(a-4)
(a+4)•(a-4)
Now, on the left hand side, the <span> (a+4) •</span> (a-4) cancels out the denominator, while, on the right hand side, zero times anything is still zero.
The equation now takes the shape :
a+16 = 0
Solving a Single Variable Equation :
<span> 6.2 </span> Solve : a+16 = 0<span>
</span>Subtract 16 from both sides of the equation :<span>
</span> a = -16
One solution was found :
<span> a = -16</span>
