Question

Решите хотя бы одно задание, буду очень благодарна!!!1. Упростите выражения:а) -2x(1-x)+(2x-3)(x-1)б) 4(1+3x)²-24xв) (x+4)(x-4)-(x-3)²2. Разложите на множители:а) y-100y³б) 7a²-14ab+7b²в) 16-y⁴3. Решите уравнение:(x³-6)(x²+2)=(x²-2)²-x4. Представьте в виде произведения:а) xy²-x+5-5y²б) m⁸+27m⁵5. Докажите, что при любых значениях x выражение принимает положительные значения:x²+8x+19

Answer 1

1) а) -2х(1-х)+(2х-3)(х-1)=-2х+2x^2+2х^2-2х-3х+3=4х^2-7х+3

б)4(1+3х)^2-24х=4(3х+1)^2-24х=4(9х^2+6х+1)-24х=36х^2+24х+4-24х=36х^2+4

в)(х+4)(х-4)-(х-3)^2=х^2-4х+4х-16-(х^2-6х+9)=х^2-4х+4х-16-х^2+6х-9=6х-25        

 

 

 

 

Answer 2

1. а) 4(х-1)(х-0,75)
б) 4+36х^2
в) 6х- 25
2. а) у(1-100у^2)
Что успела

Answer 3

Упростите выражения:

а) -2x(1-x) + (2x-3)(x-1)

To simplify this expression, we will expand the brackets and combine like terms:

-2x(1-x) + (2x-3)(x-1) = -2x + 2x² - (2x-3)(1-x) = -2x + 2x² - (2x - 3 - 2x² + 3x) = -2x + 2x² - 2x + 2x² - 3 + 3x = 4x² - 3

So, the simplified expression is 4x² - 3.

б) 4(1+3x)² - 24x

To simplify this expression, we will expand the square and combine like terms:

4(1+3x)² - 24x = 4(1 + 6x + 9x²) - 24x = 4 + 24x + 36x² - 24x = 36x² + 4

So, the simplified expression is 36x² + 4.

в) (x+4)(x-4) - (x-3)²

To simplify this expression, we will expand the brackets and combine like terms:

(x+4)(x-4) - (x-3)² = x² - 4x + 4x - 16 - (x² - 6x + 9) = x² - 16 - (x² - 6x + 9) = x² - 16 - x² + 6x - 9 = 6x - 25

So, the simplified expression is 6x - 25.

Разложите на множители:

а) y - 100y³

To factorize this expression, we can factor out the common factor y:

y - 100y³ = y(1 - 100y²)

So, the expression can be factored as y(1 - 100y²).

б) 7a² - 14ab + 7b²

To factorize this expression, we can notice that each term is a perfect square:

7a² - 14ab + 7b² = (√7a)² - 2(√7a)(√7b) + (√7b)² = (√7a - √7b)²

So, the expression can be factored as (√7a - √7b)².

в) 16 - y⁴

To factorize this expression, we can notice that it is a difference of squares:

16 - y⁴ = (4 - y²)(4 + y²)

So, the expression can be factored as (4 - y²)(4 + y²).

Решите уравнение:

(x³ - 6)(x² + 2) = (x² - 2)² - x

To solve this equation, we will expand the brackets, simplify, and solve for x:

(x³ - 6)(x² + 2) = (x² - 2)² - x x⁵ - 6x² + 2x³ - 12 = x⁴ - 4x² + 4 - x x⁵ - 6x² + 2x³ - 12 = x⁴ - 4x² - x + 4 x⁵ - 2x⁴ - 2x³ + 3x² - x - 16 = 0

Unfortunately, this equation does not have a simple closed-form solution. It may be possible to find approximate solutions using numerical methods or calculators.

Представьте в виде произведения:

а) xy² - x + 5 - 5y²

To factorize this expression, we can group the terms:

xy² - x + 5 - 5y² = x(y² - 1) - (5y² - 5) = x(y - 1)(y + 1) - 5(y - 1)(y + 1) = (y - 1)(xy - 5)(y + 1)

So, the expression can be factored as (y - 1)(xy - 5)(y + 1).

б) m⁸ + 27m⁵

To factorize this expression, we can notice that both terms have a common factor of m⁵:

m⁸ + 27m⁵ = m⁵(m³ + 27)

So, the expression can be factored as m⁵(m³ + 27).

Докажите, что при любых значениях x выражение принимает положительные значения:

x² + 8x + 19

To prove that the expression is always positive, we can use the fact that a quadratic expression in the form ax² + bx + c, where a > 0, will always have a minimum value at x = -b/(2a). In this case, a = 1, b = 8, and c = 19.

The minimum value of x² + 8x + 19 occurs at x = -8/(2*1) = -4.

Substituting this value into the expression:

(-4)² + 8(-4) + 19 = 16 - 32 + 19 = 3

Since the minimum value of the expression is 3, and it is a quadratic with a positive leading coefficient, it will always be positive for any value of x.

Therefore, the expression x² + 8x + 19 is always positive for any value of x.