if a+3b = -10 and ab = 1/9, find a²+9b²
Answers
Answer:STEP
1
:
a
Simplify —
3
Equation at the end of step
1
:
(a2) a
(((64•————)•(b2))+4)+((32•—)•b)
9 3
STEP
2
:
a2
Simplify ——
9
Equation at the end of step
2
:
a2 32ab
(((64 • ——) • b2) + 4) + ————
9 3
STEP
3
:
Equation at the end of step 3
64a2b2 32ab
(—————— + 4) + ————
9 3
STEP
4
:
Rewriting the whole as an Equivalent Fraction
4.1 Adding a whole to a fraction
Rewrite the whole as a fraction using 9 as the denominator :
4 4 • 9
4 = — = —————
1 9
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
Adding fractions that have a common denominator :
4.2 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:
64a2b2 + 4 • 9 64a2b2 + 36
—————————————— = ———————————
9 9
Equation at the end of step
4
:
(64a2b2 + 36) 32ab
————————————— + ————
9 3
STEP
5
:
STEP
6
:
Pulling out like terms :
6.1 Pull out like factors :
64a2b2 + 36 = 4 • (16a2b2 + 9)
Calculating the Least Common Multiple :
6.2 Find the Least Common Multiple
The left denominator is : 9
The right denominator is : 3
Number of times each prime factor
appears in the factorization of:
Prime
Factor Left
Denominator Right
Denominator L.C.M = Max
{Left,Right}
3 2 1 2
Product of all
Prime Factors 9 3 9
Least Common Multiple:
9
Calculating Multipliers :
6.3 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 = 3
Making Equivalent Fractions :
6.4 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. 4 • (16a2b2+9)
—————————————————— = ——————————————
L.C.M 9
R. Mult. • R. Num. 32ab • 3
—————————————————— = ————————
L.C.M 9
Adding fractions that have a common denominator :
6.5 Adding up the two equivalent fractions
4 • (16a2b2+9) + 32ab • 3 64a2b2 + 96ab + 36
————————————————————————— = ——————————————————
9 9
STEP
7
:
Pulling out like terms :
7.1 Pull out like factors :
64a2b2 + 96ab + 36 = 4 • (16a2b2 + 24ab + 9)
Trying to factor a multi variable polynomial :
7.2 Factoring 16a2b2 + 24ab + 9
Try to factor this multi-variable trinomial using trial and error
Found a factorization : (4ab + 3)•(4ab + 3)
Detecting a perfect square :
7.3 16a2b2 +24ab +9 is a perfect square
It factors into (4ab+3)•(4ab+3)
which is another way of writing (4ab+3)2
How to recognize a perfect square trinomial:
• It has three terms
• Two of its terms are perfect squares themselves
• The remaining term is twice the product of the square roots of the other two terms
Final result :
4 • (4ab + 3)2
——————————————
9
Step-by-step explanation:
Please mark me brainlist