Question

решите неравенства с помощью графика соответсвующей квадратичной функции:1) (2х-1)^2<4x+612)-3(х^2+1)≥3х-393)6х^2-5x≥-1/4x^2-14)2+1/8x^2<3x-x^2ПОЖАЛУЙСТА ПОМОГИТЕ!!!!!

Answer 1

1) (2х-1)^2<4x+61   4x^2-4x+1-4x-61<0  4(x-5)(x+3)<0   x<5 и x<-3

4x^2-8x-60=0

D=64+960=1024

x1= 8+32

          8

x1=5

x2=8-32

        8

x2=-3

2)-3(х^2+1)≥3х-39   -3x^2-3-3x+39≥0   -3(x+3)(x-4)≥0   x≥-3 и х≥4

-3x^2-3x+36=0

D=9+432=441

x1=-3+21

         -6

x1=-3

x2=-3-21

         -6

x2=4

 

Answer 2

это первое путем сокращения. можно так решать?

Answer 3

Quadratic Inequality 1: (2x-1)^2 < 4x+6

To solve the quadratic inequality (2x-1)^2 < 4x+6 using a graph, we can follow these steps:

Step 1: Rewrite the inequality in standard form: (2x-1)^2 - (4x+6) < 0

Step 2: Expand and simplify the equation: 4x^2 - 4x + 1 - 4x - 6 < 0 4x^2 - 8x - 5 < 0

Step 3: Plot the graph of the quadratic function y = 4x^2 - 8x - 5: To graph the quadratic function, we can find the vertex and the x-intercepts.

The vertex of the quadratic function can be found using the formula x = -b/2a, where a = 4 and b = -8. Plugging in these values, we get: x = -(-8)/(2*4) = 1

The x-coordinate of the vertex is 1. To find the y-coordinate, we substitute this value into the equation: y = 4(1)^2 - 8(1) - 5 = -9

So, the vertex of the quadratic function is (1, -9).

Next, we can find the x-intercepts by setting y = 0 and solving for x: 4x^2 - 8x - 5 = 0

Using the quadratic formula, x = (-b ± √(b^2 - 4ac))/(2a), where a = 4, b = -8, and c = -5, we can find the x-intercepts: x = (8 ± √((-8)^2 - 4*4*(-5)))/(2*4) x = (8 ± √(64 + 80))/8 x = (8 ± √144)/8 x = (8 ± 12)/8

So, the x-intercepts are x = 5/2 and x = -1.

Step 4: Plot the graph: Based on the vertex and the x-intercepts, we can plot the graph of the quadratic function y = 4x^2 - 8x - 5. The graph will be a downward-opening parabola passing through the points (1, -9), (5/2, 0), and (-1, 0).

Step 5: Determine the solution to the inequality: To determine the solution to the inequality (2x-1)^2 < 4x+6, we need to find the regions on the graph where the y-values are less than zero.

From the graph, we can see that the parabola is below the x-axis between the x-intercepts of 5/2 and -1. Therefore, the solution to the inequality is: -1 < x < 5/2.

Quadratic Inequality 2: -3(x^2+1) ≥ 3x-3

To solve the quadratic inequality -3(x^2+1) ≥ 3x-3 using a graph, we can follow these steps:

Step 1: Rewrite the inequality in standard form: -3x^2 - 3 ≥ 3x - 3

Step 2: Simplify the equation: -3x^2 - 3x ≥ 0

Step 3: Plot the graph of the quadratic function y = -3x^2 - 3x: To graph the quadratic function, we can find the vertex and the x-intercepts.

The vertex of the quadratic function can be found using the formula x = -b/2a, where a = -3 and b = -3. Plugging in these values, we get: x = -(-3)/(2*(-3)) = 1/2

The x-coordinate of the vertex is 1/2. To find the y-coordinate, we substitute this value into the equation: y = -3(1/2)^2 - 3(1/2) = -3/2

So, the vertex of the quadratic function is (1/2, -3/2).

Next, we can find the x-intercepts by setting y = 0 and solving for x: -3x^2 - 3x = 0 -3x(x + 1) = 0

From this equation, we can see that the x-intercepts are x = 0 and x = -1.

Step 4: Plot the graph: Based on the vertex and the x-intercepts, we can plot the graph of the quadratic function y = -3x^2 - 3x. The graph will be a downward-opening parabola passing through the points (1/2, -3/2), (0, 0), and (-1, 0).

Step 5: Determine the solution to the inequality: To determine the solution to the inequality -3(x^2+1) ≥ 3x-3, we need to find the regions on the graph where the y-values are greater than or equal to zero.

From the graph, we can see that the parabola is above or touching the x-axis. Therefore, the solution to the inequality is: x ≤ -1 or x ≥ 0.

Quadratic Inequality 3: 6x^2-5x ≥ -1/4x^2-1

To solve the quadratic inequality 6x^2-5x ≥ -1/4x^2-1, we can follow these steps:

Step 1: Rewrite the inequality in standard form: 6x^2 - 5x + 1/4x^2 + 1 ≥ 0

Step 2: Combine like terms: (24x^2 - 20x + 1 + x^2)/4 ≥ 0 (25x^2 - 20x + 1)/4 ≥ 0

Step 3: Plot the graph of the quadratic function y = (25x^2 - 20x + 1)/4: To graph the quadratic function, we can find the vertex and the x-intercepts.

The vertex of the quadratic function can be found using the formula x = -b/2a, where a = 25/4 and b = -20/4. Plugging in these values, we get: x = -(-20/4)/(2*(25/4)) = 1/2

The x-coordinate of the vertex is 1/2. To find the y-coordinate, we substitute this value into the equation: y = (25(1/2)^2 - 20(1/2) + 1)/4 = 1/4

So, the vertex of the quadratic function is (1/2, 1/4).

Next, we can find the x-intercepts by setting y = 0 and solving for x: 25x^2 - 20x + 1 = 0

Using the quadratic formula, x = (-b ± √(b^2 - 4ac))/(2a), where a = 25, b = -20, and c = 1, we can find the x-intercepts: x = (20 ± √((-20)^2 - 4*25*1))/(2*25) x = (20 ± √(400 - 100))/50 x = (20 ± √300)/50 x = (20 ± 10√3)/50 x = (2 ± √3)/5

So, the x-intercepts are x = (2 + √3)/5 and x = (2 - √3)/5.

Step 4: Plot the graph: Based on the vertex and the x-intercepts, we can plot the graph of the quadratic function y = (25x^2 - 20x + 1)/4. The graph will be a upward-opening parabola passing through the points (1/2, 1/4), ((2 + √3)/5, 0), and ((2 - √3)/5, 0).

Step 5: Determine the solution to the inequality: To determine the solution to the inequality 6x^2-5x ≥ -1/4x^2-1, we need to find the regions on the graph where the y-values are greater than or equal to zero.

From the graph, we can see that the parabola is above or touching the x-axis. Therefore, the solution to the inequality is: x ≤ (2 - √3)/5 or x ≥ (2 + √3)/5.

Quadratic Inequality 4: 2 + 1/8x^2 < 3x - x^2

To solve the quadratic inequality 2 + 1/8x^2 < 3x - x^2, we can follow these steps:

Step 1: Rewrite the inequality in standard form: x^2 - 3x + 2 + 1/8x^2 < 0

Step 2: Combine like terms: 9/8x^2 - 3x + 2 < 0

Step 3: Plot the graph of the quadratic function y = 9/8x^2 - 3x + 2: To graph the quadratic function, we can find the vertex and the x-intercepts.

The vertex of the quadratic function can be found using the formula x = -b/2a, where a = 9/8 and b = -3. Plugging in these values, we get: x = -(-