Solution - Simplification or other simple results
Step by Step Solution
Step 1 :
Equation at the end of step 1 :
(34x2 + 180x) + 100
Step 2 :
Trying to factor by splitting the middle term
2.1 Factoring 81x2+180x+100
The first term is, 81x2 its coefficient is 81 .
The middle term is, +180x its coefficient is 180 .
The last term, "the constant", is +100
Step-1 : Multiply the coefficient of the first term by the constant 81 • 100 = 8100
Step-2 : Find two factors of 8100 whose sum equals the coefficient of the middle term, which is 180 .
| -8100 | + | -1 | = | -8101 | ||
| -4050 | + | -2 | = | -4052 | ||
| -2700 | + | -3 | = | -2703 | ||
| -2025 | + | -4 | = | -2029 | ||
| -1620 | + | -5 | = | -1625 | ||
| -1350 | + | -6 | = | -1356 | ||
| -900 | + | -9 | = | -909 | ||
| -810 | + | -10 | = | -820 | ||
| -675 | + | -12 | = | -687 | ||
| -540 | + | -15 | = | -555 | ||
| -450 | + | -18 | = | -468 | ||
| -405 | + | -20 | = | -425 | ||
| -324 | + | -25 | = | -349 | ||
| -300 | + | -27 | = | -327 | ||
| -270 | + | -30 | = | -300 | ||
| -225 | + | -36 | = | -261 | ||
| -180 | + | -45 | = | -225 | ||
| -162 | + | -50 | = | -212 | ||
| -150 | + | -54 | = | -204 | ||
| -135 | + | -60 | = | -195 | ||
| -108 | + | -75 | = | -183 | ||
| -100 | + | -81 | = | -181 | ||
| -90 | + | -90 | = | -180 | ||
| -81 | + | -100 | = | -181 | ||
| -75 | + | -108 | = | -183 | ||
| -60 | + | -135 | = | -195 | ||
| -54 | + | -150 | = | -204 | ||
| -50 | + | -162 | = | -212 | ||
| -45 | + | -180 | = | -225 | ||
| -36 | + | -225 | = | -261 | ||
| -30 | + | -270 | = | -300 | ||
| -27 | + | -300 | = | -327 | ||
| -25 | + | -324 | = | -349 | ||
| -20 | + | -405 | = | -425 | ||
| -18 | + | -450 | = | -468 | ||
| -15 | + | -540 | = | -555 | ||
| -12 | + | -675 | = | -687 | ||
| -10 | + | -810 | = | -820 | ||
| -9 | + | -900 | = | -909 | ||
| -6 | + | -1350 | = | -1356 | ||
| -5 | + | -1620 | = | -1625 | ||
| -4 | + | -2025 | = | -2029 | ||
| -3 | + | -2700 | = | -2703 | ||
| -2 | + | -4050 | = | -4052 | ||
| -1 | + | -8100 | = | -8101 | ||
| 1 | + | 8100 | = | 8101 | ||
| 2 | + | 4050 | = | 4052 | ||
| 3 | + | 2700 | = | 2703 | ||
| 4 | + | 2025 | = | 2029 | ||
| 5 | + | 1620 | = | 1625 | ||
| 6 | + | 1350 | = | 1356 | ||
| 9 | + | 900 | = | 909 | ||
| 10 | + | 810 | = | 820 | ||
| 12 | + | 675 | = | 687 | ||
| 15 | + | 540 | = | 555 | ||
| 18 | + | 450 | = | 468 | ||
| 20 | + | 405 | = | 425 | ||
| 25 | + | 324 | = | 349 | ||
| 27 | + | 300 | = | 327 | ||
| 30 | + | 270 | = | 300 | ||
| 36 | + | 225 | = | 261 | ||
| 45 | + | 180 | = | 225 | ||
| 50 | + | 162 | = | 212 | ||
| 54 | + | 150 | = | 204 | ||
| 60 | + | 135 | = | 195 | ||
| 75 | + | 108 | = | 183 | ||
| 81 | + | 100 | = | 181 | ||
| 90 | + | 90 | = | 180 | That's it |
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 90 and 90
81x2 + 90x + 90x + 100
Step-4 : Add up the first 2 terms, pulling out like factors :
9x • (9x+10)
Add up the last 2 terms, pulling out common factors :
10 • (9x+10)
Step-5 : Add up the four terms of step 4 :
(9x+10) • (9x+10)
Which is the desired factorization
Multiplying Exponential Expressions :
2.2 Multiply (9x+10) by (9x+10)
The rule says : To multiply exponential expressions which have the same base, add up their exponents.
In our case, the common base is (9x+10) and the exponents are :
1 , as (9x+10) is the same number as (9x+10)1
and 1 , as (9x+10) is the same number as (9x+10)1
The product is therefore, (9x+10)(1+1) = (9x+10)2
Final result :
(9x + 10)2
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