5-2x<=9 5-2x>=0
2x>=-4 2x<=5
x>=-2 x<=2,5
[-2;2,5]
x>2,5 2x-5<=9 2x<=14 x<=7
[2,5;7]
x[-2;7]
7-(7-(-2))/2=7-4,5=2,5
ответ 2,5
Чтобы найти наименьшее значение данного выражения, выделим из него полный квадрат:
x² - 2x - 2 = x² -2x + 1 - 3 = (x - 1)² - 3
Как известно, квадрат числа принимает неотрицательные значения. Тогда наименьшее значение выражения (x - 1)² будет равно 0, а наименьшее значение выражения (x - 1)² - 3 равно -3.
Ответ: -3.
<h2>4. <u>Решить уравнение</u>.</h2>
![(3x^2 + 9x - 5) - (7x^2 - 6x + 2) = 7 - 2x - 4x^2;](https://tex.z-dn.net/?f=%283x%5E2+%2B+9x+-+5%29+-+%287x%5E2+-+6x+%2B+2%29+%3D+7+-+2x+-+4x%5E2%3B)
![3x^2 + 9x - 5 - 7x^2 + 6x - 2 = 7 - 2x - 4x^2;](https://tex.z-dn.net/?f=3x%5E2+%2B+9x+-+5+-+7x%5E2+%2B+6x+-+2+%3D+7+-+2x+-+4x%5E2%3B)
![(3x^2 - 7x^2) + (9x + 6x) - 5 - 2 = 7 - 2x - 4x^2;](https://tex.z-dn.net/?f=%283x%5E2+-+7x%5E2%29+%2B+%289x+%2B+6x%29+-+5+-+2+%3D+7+-+2x+-+4x%5E2%3B)
![-4x^2 + 15x - 7 = 7 - 2x - 4x^2;](https://tex.z-dn.net/?f=-4x%5E2+%2B+15x+-+7+%3D+7+-+2x+-+4x%5E2%3B)
![15x - 7 = 7 - 2x;](https://tex.z-dn.net/?f=15x+-+7+%3D+7+-+2x%3B)
![15x + 2x = 7 + 7;](https://tex.z-dn.net/?f=15x+%2B+2x+%3D+7+%2B+7%3B)
![17x = 14;](https://tex.z-dn.net/?f=17x+%3D+14%3B)
![x = \frac{14}{17}.](https://tex.z-dn.net/?f=x+%3D+%5Cfrac%7B14%7D%7B17%7D.)
<u>Ответ</u>:
.
<h2>5. <u>Вычислить</u>.</h2>
1) ![\frac{216^5*36^3}{6^{20}} = \frac{(6^3)^5*(6^2)^3}{6^{20}} = \frac{6^{15}*6^6}{6^{20}} = \frac{6^{15+6}}{6^{20}} = \frac{6^{21}}{6^{20}} = 6^{21-20} = 6^1 = 6.](https://tex.z-dn.net/?f=%5Cfrac%7B216%5E5%2A36%5E3%7D%7B6%5E%7B20%7D%7D+%3D+%5Cfrac%7B%286%5E3%29%5E5%2A%286%5E2%29%5E3%7D%7B6%5E%7B20%7D%7D+%3D+%5Cfrac%7B6%5E%7B15%7D%2A6%5E6%7D%7B6%5E%7B20%7D%7D+%C2%A0%3D+%5Cfrac%7B6%5E%7B15%2B6%7D%7D%7B6%5E%7B20%7D%7D+%3D+%5Cfrac%7B6%5E%7B21%7D%7D%7B6%5E%7B20%7D%7D+%3D+6%5E%7B21-20%7D+%3D+6%5E1+%3D+6.)
2) ![\frac{18^{14}}{6^{12}*3^{14}} = \frac{(2*3^2)^{14}}{(2*3)^{12}*3^{14}} = \frac{2^{14}*(3^2)^{14}}{2^{12}*3^{12}*3^{14}} = \frac{2^{14}*3^{28}}{2^{12}*3^{12+14}} = \frac{2^{14}*3^{28}}{2^{12}*3^{26}} =\\= 2^{14-12}*3^{28-26} = 2^2*3^2 = 4*9 = 36.](https://tex.z-dn.net/?f=%5Cfrac%7B18%5E%7B14%7D%7D%7B6%5E%7B12%7D%2A3%5E%7B14%7D%7D+%3D+%5Cfrac%7B%282%2A3%5E2%29%5E%7B14%7D%7D%7B%282%2A3%29%5E%7B12%7D%2A3%5E%7B14%7D%7D+%3D+%5Cfrac%7B2%5E%7B14%7D%2A%283%5E2%29%5E%7B14%7D%7D%7B2%5E%7B12%7D%2A3%5E%7B12%7D%2A3%5E%7B14%7D%7D+%3D+%5Cfrac%7B2%5E%7B14%7D%2A3%5E%7B28%7D%7D%7B2%5E%7B12%7D%2A3%5E%7B12%2B14%7D%7D+%3D+%5Cfrac%7B2%5E%7B14%7D%2A3%5E%7B28%7D%7D%7B2%5E%7B12%7D%2A3%5E%7B26%7D%7D+%3D%5C%5C%3D+2%5E%7B14-12%7D%2A3%5E%7B28-26%7D+%3D+2%5E2%2A3%5E2+%3D+4%2A9+%3D+36.)
3) ![(\frac{6}{11})^9*(1\frac{5}{6})^7 = (\frac{6}{11})^9*(\frac{11}{6})^7 = (\frac{6}{11})^9*(\frac{6}{11})^{-7 } = (\frac{6}{11})^{9+(-7)} =\\= (\frac{6}{11})^{9-7} = (\frac{6}{11})^2 = \frac{6^2}{11^2} = \frac{36}{121}.](https://tex.z-dn.net/?f=%28%5Cfrac%7B6%7D%7B11%7D%29%5E9%2A%281%5Cfrac%7B5%7D%7B6%7D%29%5E7+%3D+%28%5Cfrac%7B6%7D%7B11%7D%29%5E9%2A%28%5Cfrac%7B11%7D%7B6%7D%29%5E7+%3D+%28%5Cfrac%7B6%7D%7B11%7D%29%5E9%2A%28%5Cfrac%7B6%7D%7B11%7D%29%5E%7B-7+%7D+%3D+%28%5Cfrac%7B6%7D%7B11%7D%29%5E%7B9%2B%28-7%29%7D+%3D%5C%5C%3D+%28%5Cfrac%7B6%7D%7B11%7D%29%5E%7B9-7%7D+%3D+%28%5Cfrac%7B6%7D%7B11%7D%29%5E2+%3D+%5Cfrac%7B6%5E2%7D%7B11%5E2%7D+%3D+%5Cfrac%7B36%7D%7B121%7D.)
<u>Ответ</u>: 6; 36;
.
<h2>
6. <u>
Найти (*)</u>
.</h2>
![(5x^2 - 3xy - y^2) - (*) = x^2 + 3xy;](https://tex.z-dn.net/?f=%285x%5E2+-+3xy+-+y%5E2%29+-+%28%2A%29+%3D+x%5E2+%2B+3xy%3B)
![(*) = (5x^2 - 3xy - y^2) - (x^2 + 3xy);](https://tex.z-dn.net/?f=%28%2A%29+%3D+%285x%5E2+-+3xy+-+y%5E2%29+-+%28x%5E2+%2B+3xy%29%3B)
![(*) = 5x^2 - 3xy - y^2 - x^2 - 3xy;](https://tex.z-dn.net/?f=%28%2A%29+%3D+5x%5E2+-+3xy+-+y%5E2+-+x%5E2+-+3xy%3B)
![(*) = 4x^2 - 6xy - y^2.](https://tex.z-dn.net/?f=%28%2A%29+%3D+4x%5E2+-+6xy+-+y%5E2.)
<u>Ответ</u>: 4x² - 6xy - y².
4-4x+x²=(2-x)²
10x-25y=5(2x-5y)
-16y²-12y=-4y(4y+3)
k²-6.25=(k-2.5)(k+2.5)
b³-125=(b-5)(b²+5b+25)
6x²+3.6x=6x(x+0.6)
40c+16+25c²=(5c+4)²
c³-c⁴+2c⁵=c³(1-c+2c²)
a²-0.04=(a-0.2)(a+0.2)
2.4ab+0.16a²+9b²=(0.4a+3b)²
p³+k⁹=(p+k³)(p²-pk³+k⁶)
1-6c²+9c⁴=(1-3c²)²
18ab³-9b⁴=9b³(2a-b)
0.36m²-25n²=(0.6m-5n)(0.6m+5n)
1/27 - a³=(¹/₃ -a)(¹/₉ + (a/3) + a²)
c³+64=(c+4)(c²-4c+16)
m⁴+2m²n³+n⁶=(m²+n³)²
0.64-4k²=(0.8-2k)(0.8+2k)
3m²+9m³=3m²(1+3m)
25a²+49+70a=(5a+7)²