Question

ПОЖАЛУЙСТА помогите решить 4^(2х-3)-3*4^(х-2)-1>или=0 за ранее спасибо♥♥♥

Answer 1

4 ^ ( 2x - 3 ) - 3 * 4 ^ ( x - 2 ) - 1 ≥ 0 
4 ^ ( - 2 ) * ( 4 ^ ( 2x - 1 ) - 3 * 4 ^ x ) ≥ 1 
0,25 * 4 ^ 2x  - 3 *4 ^ x ≥ 1 : ( 1/16 ) 
0,25 * 4 ^ 2x - 3 * 4 ^ x ≥ 16 
4 ^ x = a ; 4 ^ 2x = a²
0,25a² - 3a - 16 ≥ 0 
D = 9 + 16 = 25 = 5²
a1 = ( 3 + 5 ) : 0,5 = 16 
a2 = ( 3 - 5 ) : 0,5 = - 4 ( < 0 ) 
4 ^ x ≥ 16 
4 ^ x ≥ 4 ^ 2 
x ≥ 2 
Ответ [ 2 ; + ∞ )

Answer 2

Solving the Exponential Inequality

To solve the inequality \(4^{2x-3} - 3 \cdot 4^{x-2} - 1 \geq 0\), we can use a substitution to simplify the expression. Let's solve it step by step.

Step 1: Substitution

Let's substitute \(y = 4^{x-2}\). This will simplify the inequality.

Step 2: Rewrite the Inequality

After substitution, the inequality becomes: \[y^2 - 3y - 1 \geq 0\]

Step 3: Solve the Quadratic Inequality

We can solve the quadratic inequality \(y^2 - 3y - 1 \geq 0\) by finding the roots of the quadratic equation \(y^2 - 3y - 1 = 0\).

The discriminant of the quadratic equation is given by: \[D = b^2 - 4ac\]

Where: - \(a = 1\) - \(b = -3\) - \(c = -1\)

The discriminant \(D\) is: \[D = (-3)^2 - 4(1)(-1) = 9 + 4 = 13\]

Since the discriminant \(D\) is positive, the quadratic equation has two distinct real roots.

The roots of the quadratic equation can be found using the quadratic formula: \[y = \frac{-b \pm \sqrt{D}}{2a}\]

Substitute the values of \(a\), \(b\), and \(D\) into the quadratic formula to find the roots.

Step 4: Conclusion

After finding the roots of the quadratic equation, we can determine the intervals where the inequality \(y^2 - 3y - 1 \geq 0\) is satisfied. Finally, we can substitute back \(y = 4^{x-2}\) to find the solution for the original inequality.