Find all points, if any, where y - 4x = 12 intersects 2 - y = 2(x + 2)²

Solution:

It is given that,

y - 4x = 12 intersects 2 - y = 2(x + 2)²,

Now, consider y - 4x = 12, this can also be written as below,

y = 4x + 12 … [equation 1]

Also consider the other equation,

2 - y = 2(x + 2)² this can also be written as below,

2 - 2(x+2)² = y … [equation 2]

We have both the y's isolated.

We consider, as stated earlier, that in order to intersect, the equations have equal x values and equal y values.

So we know that y = y and x = x.

Since y = y, we can say that both equations contain sides opposite y that are equal.

Then,

4x + 12 = 2 - 2(x + 2)²

Now we have to solve for x,

We know that, (a + b)² = a² + 2ab + b²

4x + 12 = 2 - 2(x² + (2 × x × 2) + 2²)

4x + 12 = 2 - 2(x² + 4x + 4)

4x + 12 = 2 -2x² - 8x - 8

By transposing we get,

2x² + 8x + 4x + 12 + 6 = 0

2x² + 12x + 18 = 0

By taking common terms outside, we get

2(x² + 6x + 9) = 0

2(x + 3)² = 0

(x + 3)² = 0/2

x + 3 = √0

x = -3

Now we have to substitute the value of x in equation 1 to get y,

y = 4(-3) + 12

y = -12 + 12

y = 0

Therefore, x = - 3 and y = 0 are the points where y - 4x = 12 intersects 2 - y = 2(x + 2)².

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Find all points, if any, where y - 4x = 12 intersects 2 - y = 2(x + 2)²

Summary:

x = - 3 and y = 0 are the points where y - 4x = 12 intersects 2 - y = 2(x + 2)².