Question

- 10 x ^ 2 + 4 x < 0

Answer 1

2*5^(2x) - 5*(2^x)*(5^x) + 2*(2^2x) = 0    /( 2^2x)
2*(5/2)^(2x) - 5*(5/2)^x + 2 = 0
(5/2)^x = z
2*(z^2) - 5z + 2 =0
D = 25 - 4*2*2 = 9
z1 = (5 - 3) /4 = 1/2
z2 = (5 + 3)/4 = 2
(5/2)^x = 1/2
x = log(5/2)  1/2
(5/2)^x = 2
x = log(5/2)  2

Answer 2

To solve this inequality, we need to find the values of x that make the expression 10x^2 + 4x less than zero.

First, we can factor out a common factor of 2x to get:

2x(5x + 2) < 0

Now we can use the product rule of inequalities to solve for the values of x that make this expression less than zero:

• If both factors are positive or both are negative, then the product is positive.
• If one factor is zero, then the product is zero.
• If one factor is positive and the other is negative, then the product is negative.

We can use this rule to create a sign chart:

scssCopy code
|     -    | 0     |     +    |
| ---------|-------|---------|
| 2x       |  -    |   0     |    +   |
| 5x + 2   |  -2/5 |  -2/5 < x < 0 |  0 < x < -2/5  |
| 2x(5x+2) |  +    |   0     |    -   |

From this chart, we can see that the expression is negative when -2/5 < x < 0. Therefore, the solution to the inequality is:

-2/5 < x < 0