the factors of x^4/4-y^2/9 are
Answers
Answer:
Step-by-step explanation:
Step 1 :
y2
Simplify ——
9
Equation at the end of step 1 :
(x2) y2
(———— - ——) - 1 = 0
4 9
Step 2 :
x2
Simplify ——
4
Equation at the end of step 2 :
x2 y2
(—— - ——) - 1 = 0
4 9
Step 3 :
Calculating the Least Common Multiple :
3.1 Find the Least Common Multiple
The left denominator is : 4
Least Common Multiple:
36
Calculating Multipliers :
3.2 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 = 9
Right_M = L.C.M / R_Deno = 4
Making Equivalent Fractions :
3.3 Rewrite the two fractions into equivalent fractions
Two fractions are called equivalent if they have the same numeric value.
For example : 1/2 and 2/4 are equivalent, y/(y+1)2 and (y2+y)/(y+1)3 are equivalent as well.
To calculate equivalent fraction , multiply the Numerator of each fraction, by its respective Multiplier.
L. Mult. • L. Num. x2 • 9
—————————————————— = ——————
L.C.M 36
R. Mult. • R. Num. y2 • 4
—————————————————— = ——————
L.C.M 36
Adding fractions that have a common denominator :
3.4 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:
x2 • 9 - (y2 • 4) 9x2 - 4y2
————————————————— = —————————
36 36
Equation at the end of step 3 :
(9x2 - 4y2)
——————————— - 1 = 0
36
Step 4 :
Rewriting the whole as an Equivalent Fraction :
4.1 Subtracting a whole from a fraction
Rewrite the whole as a fraction using 36 as the denominator :
1 1 • 36
1 = — = ——————
1 36
Equivalent fraction : The fraction thus generated looks different but has the same value as the whole
Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator
Trying to factor as a Difference of Squares :
4.2 Factoring: 9x2 - 4y2
Theory : A difference of two perfect squares, A2 - B2 can be factored into (A+B) • (A-B)
Proof : (A+B) • (A-B) =
A2 - AB + BA - B2 =
A2 - AB + AB - B2 =
A2 - B2
Note : AB = BA is the commutative property of multiplication.
Note : - AB + AB equals zero and is therefore eliminated from the expression.
Check : 9 is the square of 3
Check : 4 is the square of 2
Check : x2 is the square of x1
Check : y2 is the square of y1
Factorization is : (3x + 2y) • (3x - 2y)
Adding fractions that have a common denominator :
4.3 Adding up the two equivalent fractions
(3x+2y) • (3x-2y) - (36) 9x2 - 4y2 - 36
———————————————————————— = ——————————————
36 36
The right denominator is : 9