I could not figure out the answer to this Problem:
The quadratic function
f(x)= -2x^2 =4x +3 can be used to solve the following inequality:
-2x^2 -4x < -3
First use the Quadratic Formula to find the x-intercept of f.
This problem asks that the answer be separated by commas and rounded to nearest tenth
Also solution set is requested and put in interval notation and numbers rounded to nearest tenth
My answers to this problem where incorrect The x-intercepts I found were 2.4,5.8
and the solution set I had is: (-infinity, 2.4] union [5.8,infinity)
solving the quadratic
-2x^2 - 4x + 3 = 0 or
2x^2 + 4x - 3 = 0 yields
x = (-4 ± √40)/4
x = .58 or -2.58 not your answers.
so x < -2.58 OR x > .58
round off and express in the form they want.
So once I have to convert sqrt of 40 and solve from there?
Thank you so much for your help.
To solve the inequality -2x^2 - 4x < -3 using the quadratic function f(x) = -2x^2 + 4x + 3, you need to find the x-intercepts of the quadratic function and determine the intervals where the function is below the given inequality.
Step 1: Find the x-intercepts of the quadratic function f(x) = -2x^2 + 4x + 3 using the quadratic formula:
The quadratic formula is given by: x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = -2, b = 4, and c = 3.
Substituting these values into the quadratic formula, we get:
x = (-4 ± √(4^2 - 4(-2)(3))) / (2(-2))
x = (-4 ± √(16 + 24)) / (-4)
x = (-4 ± √40) / (-4)
x = (-4 ± 2√10) / (-4)
x = 1 ± 0.632 (- approximations)
The x-intercepts of the quadratic function are approximately 1.632 and -0.632.
Step 2: Determine the intervals where the function is below the given inequality:
To find the intervals where the function f(x) = -2x^2 + 4x + 3 is below the inequality -2x^2 - 4x < -3, we need to examine the behavior of the quadratic function.
-2x^2 - 4x < -3 can be rewritten as -2x^2 - 4x + 3 < 0.
We know that the quadratic function f(x) = -2x^2 + 4x + 3 is a downward-opening parabola since the coefficient of x^2 is negative (-2). The x-intercepts divide the number line into intervals, which we need to test to determine if the function is below zero.
Plug in test points from each interval into the quadratic function f(x) = -2x^2 + 4x + 3 and determine whether the result is less than zero.
Let's consider the intervals:
( -infinity, -0.632 )
(-0.632, 1.632)
(1.632, +infinity)
Choosing a test point from each interval, we can evaluate the sign of the function:
For the interval (-infinity, -0.632):
If we plug in x = -1, f(-1) = -2(-1)^2 + 4(-1) + 3 = 1 - 4 + 3 = 0, which is not less than zero.
For the interval (-0.632, 1.632):
If we plug in x = 0, f(0) = -2(0)^2 + 4(0) + 3 = 3, which is greater than zero.
For the interval (1.632, +infinity):
If we plug in x = 2, f(2) = -2(2)^2 + 4(2) + 3 = -8 + 8 + 3 = 3, which is not less than zero.
Based on these evaluations, we find that the function is less than zero only in the interval (-0.632, 1.632).
Step 3: Obtain the solution set in interval notation and round to the nearest tenth:
The solution set in interval notation is (-0.632, 1.632) rounded to the nearest tenth.
Therefore, the correct answer for the x-intercepts is approximately -0.6, 1.6, and the solution set in interval notation is (-0.6, 1.6) rounded to the nearest tenth.