Solution - Reducing fractions to their lowest terms
Step by Step Solution
Reformatting the input :
Changes made to your input should not affect the solution:
(1): "20.25" was replaced by "(2025/100)".
Step 1 :
81
Simplify ——
4
Equation at the end of step 1 :
81
((x2) + 9x) + ——
4
Step 2 :
Rewriting the whole as an Equivalent Fraction :
2.1 Adding a fraction to a whole
Rewrite the whole as a fraction using 4 as the denominator :
x2 + 9x (x2 + 9x) • 4
x2 + 9x = ——————— = —————————————
1 4
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
Step 3 :
Pulling out like terms :
3.1 Pull out like factors :
x2 + 9x = x • (x + 9)
Adding fractions that have a common denominator :
3.2 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:
x • (x+9) • 4 + 81 4x2 + 36x + 81
—————————————————— = ——————————————
4 4
Trying to factor by splitting the middle term
3.3 Factoring 4x2 + 36x + 81
The first term is, 4x2 its coefficient is 4 .
The middle term is, +36x its coefficient is 36 .
The last term, "the constant", is +81
Step-1 : Multiply the coefficient of the first term by the constant 4 • 81 = 324
Step-2 : Find two factors of 324 whose sum equals the coefficient of the middle term, which is 36 .
| -324 | + | -1 | = | -325 | ||
| -162 | + | -2 | = | -164 | ||
| -108 | + | -3 | = | -111 | ||
| -81 | + | -4 | = | -85 | ||
| -54 | + | -6 | = | -60 | ||
| -36 | + | -9 | = | -45 | ||
| -27 | + | -12 | = | -39 | ||
| -18 | + | -18 | = | -36 | ||
| -12 | + | -27 | = | -39 | ||
| -9 | + | -36 | = | -45 | ||
| -6 | + | -54 | = | -60 | ||
| -4 | + | -81 | = | -85 | ||
| -3 | + | -108 | = | -111 | ||
| -2 | + | -162 | = | -164 | ||
| -1 | + | -324 | = | -325 | ||
| 1 | + | 324 | = | 325 | ||
| 2 | + | 162 | = | 164 | ||
| 3 | + | 108 | = | 111 | ||
| 4 | + | 81 | = | 85 | ||
| 6 | + | 54 | = | 60 | ||
| 9 | + | 36 | = | 45 | ||
| 12 | + | 27 | = | 39 | ||
| 18 | + | 18 | = | 36 | That's it |
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 18 and 18
4x2 + 18x + 18x + 81
Step-4 : Add up the first 2 terms, pulling out like factors :
2x • (2x+9)
Add up the last 2 terms, pulling out common factors :
9 • (2x+9)
Step-5 : Add up the four terms of step 4 :
(2x+9) • (2x+9)
Which is the desired factorization
Multiplying Exponential Expressions :
3.4 Multiply (2x+9) by (2x+9)
The rule says : To multiply exponential expressions which have the same base, add up their exponents.
In our case, the common base is (2x+9) and the exponents are :
1 , as (2x+9) is the same number as (2x+9)1
and 1 , as (2x+9) is the same number as (2x+9)1
The product is therefore, (2x+9)(1+1) = (2x+9)2
Final result :
(2x + 9)2
—————————
4
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