Find the length of the segment AB if points A and B are the intersection points of the parabolas with equations y=−x2+9 and y=2x2−3 .

The distance between two points (x₁, y₁) and (x₂, y₂) is given by:

With this, we will found that the distance between A and B is: 4 units.

The first thing we need to do is find A and B.

We know that the points are the intersection between the two parabolas.

y = -x^2 + 9

y = 2*x^2 - 3

The intersection is given by the zeros of the difference:

(2*x^2 - 3) - (-x^2 + 9) = 0

3*x^2 -12 = 0

3*x^2 = 12

x^2 = 12/3 = 4

x = √4 = ±2

To find the points, we need to evaluate one of the parabolas in these two x-values.

Let's use the first one:

y = -(2)^2 + 9 = 5

So we have the point (2, 5)

For the other point:

y = -(-2)^2 + 9 = 5

So we have the point (-2, 5)

Then we can define:

A =  (2, 5)

B = (-2, 5)

Using the distance equation we get:

The distance between A and B is 4.

If you want to learn more, you can read:

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Answer:

Answer:

AB=4

Step-by-step explanation:

Answer:

AB=4

Step-by-step explanation:

1. Since you are finding the intersection points of two parabolas:

a. y=-x²+9

b. y=2x²-3

2. You have to set them equal to each other:

2x²-3= - x²+9

2x²+x=9+3

3x²=12

x²=4

This is the crucial part; the absolute value of x is equal to plus minus the square root of 4, since either -2 squared with parentheses or 2 squared is equal to 4.

√x²=±√4

x=±2

or

x=2; x=-2

3. Then you substitute them into each equation. For this step, any sign 2 will work.

a. y=-(2)²+9

y=-4+9

y=5

b. y=2(2)²-3

y=8-3

y=5

4. So our coordinates will be (2,5) and (-2,5). These are the points of intersection.

5. Now we use the distance formula:

The subscripts didn't work for this but I mean the square root of x 2 - x 1 in parentheses plus y 2 -y 1.

=

√16=

4

The absolute value rule that I mentioned above doesn't work for this because its a distance and you can't have a negative distance.

So AB=4