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Let W and Z be the points of intersection between the parabola whose graph is y = -X2-2X+3, and the line whose equation is y = X-7. What is the length of the line segment WZ?

Respuesta :

To find the length of the line segment WZ, where W and Z are the points of intersection between the parabola and the line, follow these steps:

1. Set the two equations equal to each other to find the points of intersection:

-X^2 - 2X + 3 = X - 7

2. Rearrange the equation to set it equal to zero:

-X^2 - 3X + 10 = 0

3. Solve the quadratic equation to find the x-coordinates of W and Z:

Factor the quadratic equation: -(X-2)(X-5) = 0

Set each factor equal to zero:

X - 2 = 0 => X = 2 (Point W)

X - 5 = 0 => X = 5 (Point Z)

4. Substitute the x-values into the equation y = X - 7 to find the y-coordinates:

For W (2, y): y = 2 - 7 = -5 (Point W is (2, -5))

For Z (5, y): y = 5 - 7 = -2 (Point Z is (5, -2))

5. Calculate the distance between points W and Z using the distance formula:

Length of WZ = sqrt((5-2)^2 + (-2-(-5))^2)

= sqrt(3^2 + 3^2)

= sqrt(18)

= 3√2

Therefore, the length of the line segment WZ is 3√2 units.

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