X 2 2x 3 List The Interval Endpoints Calculator
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This calculator helps determine the interval endpoints for the quadratic expression x² + 2x - 3. It identifies the critical points and divides the number line into intervals where the expression's behavior changes.
What are interval endpoints?
Interval endpoints are the critical points that divide the number line into segments where a function's behavior changes. For quadratic functions, these are typically the roots of the equation and the vertex.
In the expression x² + 2x - 3, the interval endpoints are the solutions to the equation x² + 2x - 3 = 0. These points divide the number line into intervals where the quadratic is positive, negative, or has a minimum/maximum.
How to find interval endpoints
To find the interval endpoints for x² + 2x - 3:
- Set the quadratic equation equal to zero: x² + 2x - 3 = 0
- Solve for x using the quadratic formula: x = [-b ± √(b² - 4ac)] / (2a)
- Identify the critical points and order them from least to greatest
- These points divide the number line into intervals
Quadratic Formula
For a quadratic equation ax² + bx + c = 0, the solutions are:
x = [-b ± √(b² - 4ac)] / (2a)
For our expression x² + 2x - 3, the coefficients are:
- a = 1
- b = 2
- c = -3
Example calculation
Let's calculate the interval endpoints for x² + 2x - 3:
- Set the equation to zero: x² + 2x - 3 = 0
- Apply the quadratic formula:
x = [-2 ± √(2² - 4(1)(-3))] / (2*1)
x = [-2 ± √(4 + 12)] / 2
x = [-2 ± √16] / 2
x = [-2 ± 4] / 2
- Calculate the two solutions:
x₁ = (-2 + 4)/2 = 2/2 = 1
x₂ = (-2 - 4)/2 = -6/2 = -3
- The interval endpoints are x = -3 and x = 1
Note
The interval endpoints divide the number line into three intervals:
- (-∞, -3)
- (-3, 1)
- (1, ∞)
FAQ
What are interval endpoints used for?
Interval endpoints help analyze the behavior of functions by identifying where the function changes its sign or reaches its maximum/minimum points.
How do I know if I've found all interval endpoints?
For quadratic equations, you should always find two real roots (unless the discriminant is zero). These roots are your interval endpoints.
Can interval endpoints be negative?
Yes, interval endpoints can be negative. In our example, one endpoint is -3, which is negative.