Please help with geometry problems
1 answer:
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Answer:
1. (4a+3)(4a-3)
2. (2m + 5)(2m-5)
3. (4b-5)²
4. (2x-1)²
5. (3x+1)(3x-1)
6. (n+5)(n-5)
7. (n²-10)²
8. (a²+3)(a²-3)
9. (k²+6)(k²-6)
10. (n²+7)(n²-7)
Step-by-step explanation:
For special products like these, just remember the
Difference of Squares formula:
and
Square of difference formula:
1. 16a²-9 follows the difference of squares form:
a² - b² = (a+b)(a-b)
So all we need to do is get the square root of a and b and follow the form:
(√16a²) - (√9) = (4a +3) (4a - 3)
Let's check through FOIL:
(4a +3) (4a - 3)
16a² - 12a + 12a - 9
16a² - 9
2. 4m² - 25 also follows the same form of difference of squares:
(√4m²) - (√25) = (2m +5) (2m - 5)
Let's check!
(2m +5) (2m - 5)
4m² + 10m - 10m -25
4m² - 25
3. 16b² - 40b + 25 follows the form of square of difference:
a² - 2ab + b² = (a - b)²
√16b² = 4b
√25 = 5
(4b - 5)²
Let's check!
(4b-5)(4b - 5)
16b² - 20b - 20b + 25
16b² - 40b + 25
HEre's another way to look at it:
First multiply the coefficients of the first and last term:
16 x 25 = 400
Next think of two numbers that when you multiply them, you will come up with 400 and sum up to -40. The two numbers would be -20 and -20 which will be the middle term
So it would fill in this way:
16b² - 20b - 20b + 25
Factor the equation:
(16b² - 20b) (-20b + 25)
4b(4b-5) - 5(4b-5)
(4b-5)(4b-5) = (4b-5)²
4. 4x² - 4x + 1 follows that same form:
√4x² = 2x
√1 = 1
(2x - 1)²
Let's check!
(2x - 1)(2x - 1)
4x² - 2x - 2x + 1
4x²- 4x + 1
5. 9x² - 1 follows the form of difference of squares:
√9x² - √1 = (3x + 1)(3x - 1)
Let's check!
(3x + 1)(3x - 1)
9x² + 3x - 3x - 1
9x² - 1
6. n² - 25 follows the form of difference of squares:
√n² - √25 = (n + 5)(n - 5)
Let's check!
(n + 5)(n - 5)
n² + 5n - 5n - 25
n² - 25
7. n⁴ - 100
√n⁴ - √100 = (n² + 10)(n² - 100)
Let's check!
(n² + 10)(n² - 100)
n⁴ + 10n² - 10n² -100
n⁴ - 100
8. a⁴ - 9
√a⁴ - √9 = (a² + 3)(a² - 3)
Let's check!
(a² + 3)(a² - 3)
a⁴ + 3a² - 3a² - 9
a⁴ - 9
9. k⁴ - 36
√k⁴ - √36 = (k² + 6) (k² - 6)
Let's check!
(k² + 6) (k² - 6)
k⁴ + 6k² - 6k² - 36
k⁴ - 36
10. n⁴ - 49
√n⁴ - √49 = (n² + 7) (n² - 7)
Let's check!
(n² + 7) (n² - 7)
n⁴ +7n² - 7n²- 49
n⁴ - 49