Question

1)выполните действия,используя формулы сокращенного умножения: (2a-3a)(2a+3)²; (m-2n)² 2)заполните пропуски: (3x + ?)²= ?+?+16 3)разложите на множители: (5а + 7)²-4b²4)найдите площадь квадрата,сторона которого равна: (2√27+√3) см.5)запишите в виде квадрата разности: 19-8√3

Answer 1

$1) (2a-3a)(2a+3)^3=(-a)(4a^2+6a+9)=-4a^3-6a^2-9a\\(m-2n)^2=m^2-4mn+4n^2\\\\2) (3x+4)^2=9x^2+24x+16\\\\3) (5a+7)^2-4b^2=(5a+7)^2-(2b)^2=(5a+7-2b)(5a+7+2b)\\\\4) S=a^2=(2\sqrt{27}+\sqrt{3})^2=4*27+4\sqrt{81}+3=108+36+3=147\\\\5) 19-8\sqrt{3}=(4-\sqrt{3})^2$

Answer 2

1. Выполним действия, используя формулы сокращенного умножения:

a) (2a - 3a)(2a + 3)²

Step 1: (2a - 3a) = -a Step 2: (2a + 3)² = (2a + 3)(2a + 3) = 4a² + 12a + 9

Now, the expression becomes: -a * (4a² + 12a + 9)

Step 3: Distribute the multiplication:

-(4a³ + 12a² + 9a)

b) (m - 2n)²

Step 1: (m - 2n)² = (m - 2n)(m - 2n)

Step 2: Distribute the multiplication:

(m * m) + (m * -2n) + (-2n * m) + (-2n * -2n)

m² - 2mn - 2mn + 4n²

Simplify the middle terms:

m² - 4mn + 4n²

2. Заполним пропуски:

(3x + ?)² = ? + ? + 16

To get the square of (3x + ?), we can observe that the middle term in the expansion of (3x + ?)² will be 2 times the product of the two unknowns. This means 2 * (3x) * (?) = 2 * 3x * ? = 6x?

Now, we need to find two numbers whose product is 6x and whose sum is 16.

The numbers are 6x and 6x because (6x) * (6x) = 36x², and 2 * 6x * 6x = 72x². So, the middle term in the expansion of (3x + 6x)² will be 72x².

Now, let's fill in the blanks:

(3x + 6x)² = (9x)² = 81x²

3. Разложим на множители:

(5а + 7)² - 4b²

To factor this expression, we first identify it as a difference of squares:

(5а + 7)² - (2b)²

The difference of squares formula is: a² - b² = (a + b)(a - b)

So, applying the formula:

(5а + 7)² - 4b² = [(5а + 7) + 2b][(5а + 7) - 2b]

Result:

(5а + 7 + 2b)(5а + 7 - 2b)

4. Найдем площадь квадрата, сторона которого равна (2√27 + √3) см.

The area of a square is given by the formula: Area = side²

Given that the side of the square is (2√27 + √3) cm, we can find its area as follows:

Area = (2√27 + √3)² Area = (2√27 + √3)(2√27 + √3) Area = 4 * 27 + 2 * 2 * √27 * √3 + 3 Area = 108 + 12√3 + 3 Area = 111 + 12√3 square centimeters

5. Запишем в виде квадрата разности: 19 - 8√3

To write the expression in the form of a square of a binomial, we need to find two numbers whose product is 8√3, and whose sum is -19.

The two numbers are -√3 and -8√3, because (-√3) * (-8√3) = 8√3 * √3 = 8√(3 * 3) = 8√9 = 8 * 3 = 24, and (-√3) + (-8√3) = -9√3.

Now, we can rewrite the expression:

19 - 8√3 = (-√3)² - 2 * (-√3) * (-8√3) + (-8√3)² = 3 + 16√3 + 64 * 3 = 3 + 16√3 + 192 = 195 + 16√3

So, 19 - 8√3 can be written as a square of the difference -√3 and 8√3:

(√3 - 8√3)² = 195 + 16√3.