What is the equation of a parabola that passes through the points (−5, −10), (−3, 2), and (2, −3)? A. y = −x2 − 2x + 5
B. y = −x2 + 6x − 7
C. y = −189x2 − 289x + 353
D. y = −x2 − 3x + 2

Here we want to find the equation of the parabola that passes through the points (-5, -10), (-3, 2), and (2, -3).

We will get, by brute force, that the correct option is A:

y = -x^2 - 2x + 5

To find the correct option we can use brute force.

We do know the options, so what we can do is just evaluate the given options in the x-values of the points, and see if we get the same y-value that the point.

We can start with option A:

A: y = -x^2 - 2x + 5

For the first point (-5, -10) we have x = -5, evaluating in this we get:

y = -(-5)^2 - 2*(-5) + 5 = -25 + 10 + 5 = -10

Now let's try with the next point (-3, 2), we have x = -3

y = -(-3)^2 - 2*(-3) + 5 = -9 + 6 + 5 = 2

Now we can try with the last point (2, -3), where x= 2

y = -(2)^2 - 2*2 + 5 = -4 - 4 + 5 = -3

So we got all the correct y-values (the second value for each point) thus we can conclude that this one is the correct option.

The correct option is A: y = -x^2 - 2x + 5

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A

Step-by-step explanation:

Using the standard form of the parabola , y = ax² + bx + c ( a ≠ 0 )

Substitute the given points into the equation and solve for a, b, c using simultaneous equations.

(- 5, 10 )

- 10 = a(- 5)² - 5b + c, that is

25a - 5b + c = - 10 → (1)

(- 3, 2 )

2 = a(- 3)² - 3b + c, that is

9a - 3b + c = 2 → (2)

(2, - 3 )

- 3 = a(2)² + 2b + c , that is

4a + 2b + c = - 3 → (3)

Eliminate c from the equations

Subtract (2) from (1) term by term

16a - 2b = - 12 → (4)

Subtract (3) from (2) term by term

5a - 5b = 5 → (5)

Multiply (4) by 5 and (5) by - 2 then add to eliminate b

80a - 10b = - 60 → (6)

- 10a + 10b = - 10 → (7)

Add (6) and (7) term by term

70a = - 70 ( divide both sides by 70 )

a = - 1

Substitute a = - 1 into (4) and evaluate for b

- 16 - 2b = - 12 ( add 16 to both sides )

- 2b = 4 ( divide both sides by - 2 )

b = - 2

Substitute a = - 1, b = - 2 into (3) and evaluate for c

- 4 - 4 + c = - 3

- 8 + c = - 3 ( add 8 to both sides )

c = 5

Thus a = - 1, b = - 2 and c = 5

Substitute these values into the standard parabola, that is

y = - x² - 2x + 5 → A