Solution - Simplification or other simple results
Step by Step Solution
Step 1 :
Equation at the end of step 1 :
(52x2 + 80x) + 64
Step 2 :
Trying to factor by splitting the middle term
2.1 Factoring 25x2+80x+64
The first term is, 25x2 its coefficient is 25 .
The middle term is, +80x its coefficient is 80 .
The last term, "the constant", is +64
Step-1 : Multiply the coefficient of the first term by the constant 25 • 64 = 1600
Step-2 : Find two factors of 1600 whose sum equals the coefficient of the middle term, which is 80 .
| -1600 | + | -1 | = | -1601 | ||
| -800 | + | -2 | = | -802 | ||
| -400 | + | -4 | = | -404 | ||
| -320 | + | -5 | = | -325 | ||
| -200 | + | -8 | = | -208 | ||
| -160 | + | -10 | = | -170 | ||
| -100 | + | -16 | = | -116 | ||
| -80 | + | -20 | = | -100 | ||
| -64 | + | -25 | = | -89 | ||
| -50 | + | -32 | = | -82 | ||
| -40 | + | -40 | = | -80 | ||
| -32 | + | -50 | = | -82 | ||
| -25 | + | -64 | = | -89 | ||
| -20 | + | -80 | = | -100 | ||
| -16 | + | -100 | = | -116 | ||
| -10 | + | -160 | = | -170 | ||
| -8 | + | -200 | = | -208 | ||
| -5 | + | -320 | = | -325 | ||
| -4 | + | -400 | = | -404 | ||
| -2 | + | -800 | = | -802 | ||
| -1 | + | -1600 | = | -1601 | ||
| 1 | + | 1600 | = | 1601 | ||
| 2 | + | 800 | = | 802 | ||
| 4 | + | 400 | = | 404 | ||
| 5 | + | 320 | = | 325 | ||
| 8 | + | 200 | = | 208 | ||
| 10 | + | 160 | = | 170 | ||
| 16 | + | 100 | = | 116 | ||
| 20 | + | 80 | = | 100 | ||
| 25 | + | 64 | = | 89 | ||
| 32 | + | 50 | = | 82 | ||
| 40 | + | 40 | = | 80 | That's it |
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 40 and 40
25x2 + 40x + 40x + 64
Step-4 : Add up the first 2 terms, pulling out like factors :
5x • (5x+8)
Add up the last 2 terms, pulling out common factors :
8 • (5x+8)
Step-5 : Add up the four terms of step 4 :
(5x+8) • (5x+8)
Which is the desired factorization
Multiplying Exponential Expressions :
2.2 Multiply (5x+8) by (5x+8)
The rule says : To multiply exponential expressions which have the same base, add up their exponents.
In our case, the common base is (5x+8) and the exponents are :
1 , as (5x+8) is the same number as (5x+8)1
and 1 , as (5x+8) is the same number as (5x+8)1
The product is therefore, (5x+8)(1+1) = (5x+8)2
Final result :
(5x + 8)2
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