Question

Find the solution for the quadratic inequality using the quadratic formula. Use -3, 0, 4 as test values.

2x^2-4x-6<0

Answer 1

To solve the inequality 2x²-4x-6<0, apply the quadratic formula to find the corresponding equation's roots, which are x = 3 and x = -1.

The solution set for the inequality is the interval (-1, 3), which can be verified by testing the values -3, 0, and 4 in the original inequality.

Step-by-step explanation:

To find the solution for the quadratic inequality 2x²-4x-6<0 using the quadratic formula, we first need to solve the corresponding quadratic equation 2x²-4x-6=0.

The quadratic formula to find the roots of any quadratic equation ax²+bx+c = 0 is given by -b ± √b² - 4ac / 2a.

In our case, a=2, b=-4, and c=-6.

Applying the values into the quadratic formula, we get the roots:

• x = (4 ± √((-4)² - 4*2*(-6))) / (2*2)
• x = (4 ± √(16 + 48)) / 4
• x = (4 ± √64) / 4
• x = (4 ± 8) / 4
• x = 3 or x = -1

The inequality 2x^2-4x-6<0 will be true where the values of x are between the roots because the coefficient of x² is positive, indicating the parabola is open upwards and it will be less than 0 in between the roots. Now, we test the values -3, 0, 4:

• For x=-3: 2(-3)^2-4(-3)-6 = 18+12-6 > 0 (not in the solution set)
• For x=0: 2(0)^2-4(0)-6 = -6 < 0 (in the solution set)
• For x=4: 2(4)^2-4(4)-6 = 32-16-6 > 0 (not in the solution set)

The solution set for the inequality is thus the interval (-1, 3).

Answer 2

To solve the quadratic inequality 2x^2 - 4x - 6 < 0 using the quadratic formula, we first need to find the roots of the quadratic equation 2x^2 - 4x - 6 = 0. The quadratic formula is given by:

x = (-b ± √(b^2 - 4ac)) / (2a)

For the given quadratic equation, a = 2, b = -4, and c = -6. Substituting these values into the quadratic formula, we get:

x = (-(-4) ± √((-4)^2 - 4 * 2 * -6)) / (2 * 2)
x = (4 ± √(16 + 48)) / 4
x = (4 ± √64) / 4
x = (4 ± 8) / 4

Simplifying further, we have:

x = (4 + 8) / 4 = 3
x = (4 - 8) / 4 = -1/2

So, the roots of the quadratic equation are x = 3 and x = -1/2.

To determine the solution to the quadratic inequality 2x^2 - 4x - 6 < 0, we can use the test values -3, 0, and 4. Substituting these values into the inequality, we get:

2(-3)^2 - 4(-3) - 6 < 0
2(9) + 12 - 6 < 0
18 + 12 - 6 < 0
24 < 0 (False)

2(0)^2 - 4(0) - 6 < 0
0 + 0 - 6 < 0
-6 < 0 (True)

2(4)^2 - 4(4) - 6 < 0
2(16) - 16 - 6 < 0
32 - 16 - 6 < 0
10 < 0 (False)

Based on the test values, we can see that the inequality is only true when x is between -1/2 and 3. Therefore, the solution to the quadratic inequality 2x^2 - 4x - 6 < 0 is -1/2 < x < 3.
Hope this helps, and you understand this, thanks for the points.