I have trouble solving quadratic inequalities. I don't understand when you use (or) and when you just put it all together.
Ex.
-2x^2+3x+6 greater than or equal to 2. How do I know which it is?
sorry, greater than or equal to 0
I NEED HELP ON FINALS FOR GRAPHING
Solving quadratic inequalities can be tricky, but I can help you understand the process!
To solve a quadratic inequality like -2x^2 + 3x + 6 ≥ 2, there are a few steps you should follow:
Step 1: Begin by setting the inequality to zero. In this case, we have -2x^2 + 3x + 6 - 2 ≥ 0, which simplifies to -2x^2 + 3x + 4 ≥ 0.
Step 2: Next, find the x-intercepts (also known as the roots or zeros) of the quadratic equation. To do this, factor the quadratic equation or use the quadratic formula. In this case, the quadratic equation cannot be easily factored, so we'll use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a.
For -2x^2 + 3x + 4 = 0, the values of a, b, and c are -2, 3, and 4, respectively. Plugging these values into the quadratic formula, we find the x-intercepts: x = (-(3) ± √((3)^2 - 4(-2)(4))) / (2(-2)). Simplifying this gives us x = (3 ± √(9 + 32)) / -4, which further simplifies to x = (3 ± √41) / -4. Therefore, the x-intercepts are approximately x ≈ -1.15 and x ≈ 1.65.
Step 3: Now, we can plot the x-intercepts on a number line. The x-intercepts divide the number line into three intervals: (-∞, -1.15), (-1.15, 1.65), and (1.65, +∞).
Step 4: Choose a test point within each interval and substitute it back into the inequality. You can use any value within each interval. To simplify the calculations, it is often advisable to choose integers. For example, we can choose -2, 0, and 2 as test points.
Step 5: Substitute the test points back into the original inequality -2x^2 + 3x + 4 ≥ 0 and check whether each point satisfies the inequality. If the inequality is satisfied, that interval is part of the solution.
For example, when we substitute the test point -2 into the inequality -2(-2)^2 + 3(-2) + 4 ≥ 0, we have 20 ≥ 0, which is true. Therefore, the interval (-∞, -1.15) is part of the solution.
Similarly, when we substitute the test point 0 into the inequality -2(0)^2 + 3(0) + 4 ≥ 0, we have 4 ≥ 0, which is also true. Therefore, the interval (-1.15, 1.65) is also part of the solution.
However, when we substitute the test point 2 into the inequality -2(2)^2 + 3(2) + 4 ≥ 0, we have -2 ≥ 0, which is false. Therefore, the interval (1.65, +∞) is not part of the solution.
Step 6: Finally, express the solution as an inequality. Since the test points -2 and 0 satisfy the original inequality, we can write the solution as (-∞, -1.15] ∪ [-1.15, 1.65].
To summarize, the solution to the quadratic inequality -2x^2 + 3x + 6 ≥ 2 is (-∞, -1.15] ∪ [-1.15, 1.65].