The solution set of 2x^2+6x+4<0
To solve the inequality 2x^2+6x+4<0, we can follow these steps:
1. Divide both sides by 2 to simplify the coefficients:
x^2 + 3x + 2 < 0
2. Factor the quadratic expression:
(x + 1)(x + 2) < 0
3. Find the critical values of x where the expression changes signs:
x + 1 = 0 ⇒ x = -1
x + 2 = 0 ⇒ x = -2
4. Create a sign chart and test the expression in each interval:
- + -
|____|____|____|
-2 -1 0
For x < -2, both factors are negative, so their product is positive.
For -2 < x < -1, only (x+2) is positive, so the product is negative.
For x > -1, both factors are positive, so their product is positive.
5. Write the solution set by combining the intervals where the expression is negative:
x ∈ (-2, -1)
Therefore, the solution set of 2x^2+6x+4<0 is the open interval (-2, -1).
2x^2+6x+4<0
x^2 + 3x + 2 < 0
(x+2)(x+1) < 0
-2 < x < -1
You are correct! The solution set of the inequality (x+2)(x+1) < 0 is (-2, -1), which means that -2 < x < -1 is the set of values of x that make the inequality true.
To find the solution set of the inequality 2x^2 + 6x + 4 < 0, follow these steps:
Step 1: Start by factoring the quadratic equation. In this case, since the coefficient of x^2 is non-zero, we can't factor it easily. Instead, we will use the quadratic formula.
Step 2: The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = 2, b = 6, and c = 4. Substituting these values into the quadratic formula:
x = (-6 ± √(6^2 - 4(2)(4))) / (2(2))
x = (-6 ± √(36 - 32)) / 4
x = (-6 ± √(4)) / 4
x = (-6 ± 2) / 4
Step 3: Simplify the expression:
x = (-6 + 2) / 4 or x = (-6 - 2) / 4
x = -4/4 or x = -8/4
x = -1 or x = -2
Step 4: Now, we have the critical points -1 and -2. We can use these points to determine the solution set for the inequality.
Step 5: Choose a test point in each of the three intervals: (-∞, -2), (-2, -1), and (-1, ∞).
Step 6: Substitute the test points into the original inequality to determine if they satisfy it.
For instance, let's choose x = -3 as a test point in the interval (-∞, -2).
Substituting x = -3 into the inequality:
2(-3)^2 + 6(-3) + 4 < 0
2(9) - 18 + 4 < 0
18 - 18 + 4 < 0
4 < 0
Since 4 is not less than 0, the test point x = -3 does not satisfy the inequality.
Step 7: Repeat step 6 for the test points in the other intervals, (-2, -1) and (-1, ∞).
For the interval (-2, -1), let's choose x = -1.5 as a test point.
Substituting x = -1.5 into the inequality:
2(-1.5)^2 + 6(-1.5) + 4 < 0
2(2.25) - 9 + 4 < 0
4.5 - 9 + 4 < 0
-0.5 < 0
Since -0.5 is less than 0, the test point x = -1.5 satisfies the inequality.
For the interval (-1, ∞), let's choose x = 0 as a test point.
Substituting x = 0 into the inequality:
2(0)^2 + 6(0) + 4 < 0
0 + 0 + 4 < 0
4 < 0
Since 4 is not less than 0, the test point x = 0 does not satisfy the inequality.
Step 8: Based on the results of the test points, the solution set for the inequality 2x^2 + 6x + 4 < 0 is:
x ∈ (-2, -1)
To find the solution set of the quadratic inequality 2x^2 + 6x + 4 < 0, we can follow these steps:
Step 1: Solve the equation 2x^2 + 6x + 4 = 0.
To do this, we can use the quadratic formula: x = (-b ± √(b^2 - 4ac))/(2a).
In this case, a = 2, b = 6, and c = 4.
Plugging these values into the quadratic formula, we can calculate the solutions.
x = (-6 ± √(6^2 - 4(2)(4)))/(2(2))
x = (-6 ± √(36 - 32))/(4)
x = (-6 ± √(4))/(4)
x = (-6 ± 2)/(4)
So the two solutions are x = (-6 + 2)/4 = -4/4 = -1 and x = (-6 - 2)/4 = -8/4 = -2.
Step 2: Determine the critical points.
The critical points are the solutions we found in step 1, which are -1 and -2.
Step 3: Plot these critical points on a number line.
On a number line, we mark -1 and -2.
Step 4: Choose test values.
Select test values for each of the intervals created on the number line. Since our inequality is strict (<), we need to choose values such that they make the inequality true or false.
For the interval (-∞, -2), we can choose x = -3.
For the interval (-2, -1), we can choose x = 0.
For the interval (-1, +∞), we can choose x = 1.
Step 5: Test the inequalities.
Substitute the test values into the original inequality and check if they satisfy the inequality.
For x = -3:
2(-3)^2 + 6(-3) + 4 < 0
18 - 18 + 4 < 0
4 < 0 (False)
For x = 0:
2(0)^2 + 6(0) + 4 < 0
4 < 0 (False)
For x = 1:
2(1)^2 + 6(1) + 4 < 0
2 + 6 + 4 < 0
12 < 0 (False)
Step 6: Identify the solution set.
Based on the testing, we see that none of the test values satisfy the inequality. Therefore, there are no values of x that make the inequality 2x^2 + 6x + 4 < 0 true.
Therefore, the solution set of the inequality 2x^2 + 6x + 4 < 0 is an empty set, or in interval notation, (-∞, ∞).