Question

Y = −x2 + 4x + 12 and y = 9x + 16

Answer 1

To find the point(s) of intersection between the two equations y = -x^2 + 4x + 12 and y = 9x + 16, we can set the two equations equal to each other and solve for x:

-x^2 + 4x + 12 = 9x + 16

Rearranging terms, we get:

-x^2 - 5x - 4 = 0

We can solve this quadratic equation by factoring:

-x^2 - 5x - 4 = -(x+1)(x+4) = 0

Therefore, either x+1=0 or x+4=0, which gives us x=-1 or x=-4.

To find the corresponding y-values, we can substitute these values of x back into either of the original equations:

For x=-1, y = -(-1)^2 + 4(-1) + 12 = 9

So, one point of intersection is (-1, 9).

For x=-4, y = -(-4)^2 + 4(-4) + 12 = -4

So, the other point of intersection is (-4, -4).

Therefore, the two equations intersect at the points (-1, 9) and (-4, -4)

Answer 2

To solve the system of equations Y = −x^2 + 4x + 12 and y = 9x + 16, we can substitute y in the first equation with 9x + 16 from the second equation.

Y = −x^2 + 4x + 12

y = 9x + 16

Substitute y = 9x + 16 in the first equation:

Y = −x^2 + 4x + 12

9x + 16 = −x^2 + 4x + 12

Rearrange the equation to get it in standard quadratic form:

x^2 - 5x + 4 = 0

Factor the quadratic:

(x - 1)(x - 4) = 0

So, x = 1 or x = 4.

Substitute these values of x back into one of the equations to find the corresponding values of y.

For x = 1:

Y = −1^2 + 4(1) + 12 = 15

y = 9(1) + 16 = 25

So, one solution is (1, 15) and (1, 25).

For x = 4:

Y = −4^2 + 4(4) + 12 = 12

y = 9(4) + 16 = 52

So, the other solution is (4, 12) and (4, 52).

Therefore, the system of equations has two solutions: (1, 15) and (4, 12).

Explanation: