Question

40x-16x^2-25>=0 плиз очень надо

Answer 1

$\displaystyle 40x-16x^2-25\geq 0\\-(16x^2-40x+25)\geq 0\\-(4x-5)^2\geq 0\\(4x-5)^2\leq 0\\(4x-5)^2=0\\\\x=\frac{5}{4}=1\frac{1}{4}=1,25$

Answer 2

To solve the inequality 40x - 16x^2 - 25 >= 0, we can follow these steps:

1. Rearrange the inequality: Move all terms to one side of the inequality sign to obtain a quadratic expression. In this case, we have 40x - 16x^2 - 25 >= 0 becomes -16x^2 + 40x - 25 >= 0.

2. Factor the quadratic expression: To factor the quadratic expression, we need to find two binomials that multiply to give the quadratic expression. In this case, we have -16x^2 + 40x - 25. Factoring this expression may not be straightforward, so we can use the quadratic formula to find the roots of the quadratic equation -16x^2 + 40x - 25 = 0.

3. Apply the quadratic formula: The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x can be found using the formula x = (-b ± √(b^2 - 4ac)) / (2a). In our case, a = -16, b = 40, and c = -25. Plugging these values into the quadratic formula, we can find the roots of the equation.

4. Determine the critical points: The critical points are the x-values where the quadratic expression equals zero. These points divide the number line into intervals.

5. Test the intervals: Choose a test point from each interval and substitute it into the quadratic expression. If the result is greater than or equal to zero, then that interval satisfies the inequality. If the result is less than zero, then that interval does not satisfy the inequality.

6. Write the solution: Combine the intervals that satisfy the inequality to write the final solution.

Let's go through these steps in detail.

Step 1: Rearrange the inequality

The inequality 40x - 16x^2 - 25 >= 0 can be rearranged as -16x^2 + 40x - 25 >= 0.

Step 2: Factor the quadratic expression

Factoring the quadratic expression -16x^2 + 40x - 25 may not be straightforward, so let's use the quadratic formula to find the roots of the equation.

Step 3: Apply the quadratic formula

The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x can be found using the formula x = (-b ± √(b^2 - 4ac)) / (2a).

In our case, a = -16, b = 40, and c = -25. Plugging these values into the quadratic formula, we get:

x = (-40 ± √(40^2 - 4(-16)(-25))) / (2(-16))

Simplifying further:

x = (-40 ± √(1600 - 1600)) / (-32)

x = (-40 ± √0) / (-32)

x = -40 / (-32)

x = 5/4

So, the quadratic equation -16x^2 + 40x - 25 = 0 has a double root at x = 5/4.

Step 4: Determine the critical points

The critical points are the x-values where the quadratic expression equals zero. In our case, the only critical point is x = 5/4.

Step 5: Test the intervals

To determine which intervals satisfy the inequality, we can choose test points from each interval and substitute them into the quadratic expression.

Let's consider the intervals: - Interval 1: x < 5/4 - Interval 2: x > 5/4

For Interval 1, let's choose x = 0 as the test point: - Substituting x = 0 into the quadratic expression -16x^2 + 40x - 25, we get -25. Since -25 is less than zero, Interval 1 does not satisfy the inequality.

For Interval 2, let's choose x = 2 as the test point: - Substituting x = 2 into the quadratic expression -16x^2 + 40x - 25, we get 7. Since 7 is greater than zero, Interval 2 satisfies the inequality.

Step 6: Write the solution

Combining the intervals that satisfy the inequality, we can write the final solution as:

x > 5/4

Therefore, the solution to the inequality 40x - 16x^2 - 25 >= 0 is x > 5/4.

Please note that this solution assumes that we are working with real numbers.