Intersection Points & Finding Unknown Variable

[confusedatmath]

The line with equation y = x + k, where k is a real number, intersects the parabola with equation y = x^2 + x − 2 in two distinct points if

I first made the equations equal each other

x + k = x^2 + x − 2
0 = x^2 -2 -k

From here i thought you use the discriminate a=1 b=o c=-2-k

but this isn't right, because the answer to choose from

k < − 2

k > − 2

k = − 2

k < 2

k ≠ 2

 

[mente oscura]

The line with equation y = x + k, where k is a real number, intersects the parabola with equation y = + x − 2 in two distinct points if

I first made the equations equal each other

x + k = + x − 2
0 = -2 -k

From here i thought you use the discriminate a=1 b=o c=-2-k

but this isn't right, because the answer to choose from

k < − 2

k > − 2

k = − 2

k < 2

k ≠ 2

Hello.

Check the wording of the question. The parable, need that 'x' is not high to the power 1. (would be a straight line)

Regards.

 

[confusedatmath]

Hello.

Check the wording of the question. The parable, need that 'x' is not high to the power 1. (would be a straight line)

Regards.

i fixed it :p read again, it was a mistake i forgot the x^2

 

[mente oscura]

The line with equation y = x + k, where k is a real number, intersects the parabola with equation y = x^2 + x − 2 in two distinct points if

I first made the equations equal each other

x + k = x^2 + x − 2
0 = x^2 -2 -k

From here i thought you use the discriminate a=1 b=o c=-2-k

but this isn't right, because the answer to choose from

k < − 2

k > − 2

k = − 2

k < 2

k ≠ 2

Now yes.

$$0=x^2-2-k$$

$$x^2=k+2$$

$$x=\pm{} \sqrt{k+2}$$

1º) $$k<-2 \rightarrow{}x \cancel{\in}{R}$$

2º) $$k>-2 \rightarrow{}x \in{R}$$

3º) $$k=-2 \rightarrow{}x=0$$, only a breakpoint.

Regards.

 

[CellCoree]

I can provide a mathematical explanation for the intersection points and finding the unknown variable in this scenario. Firstly, it is important to understand that the line and the parabola intersect at points where their y-coordinates are equal. This can be represented as:

x + k = x^2 + x − 2

Simplifying the equation, we get:

x^2 + x − 2 − (x + k) = 0

x^2 + x − x − 2 − k = 0

x^2 − k − 2 = 0

This is a quadratic equation in terms of x, which can be solved using the quadratic formula:

x = [-b ± √(b^2 - 4ac)]/2a

In this case, a = 1, b = -1, and c = -k - 2. Substituting these values in the formula, we get:

x = [1 ± √(1 - 4(-k-2))]/2

x = [1 ± √(1 + 4k + 8)]/2

x = [1 ± √(4k + 9)]/2

Now, for the line and the parabola to intersect at two distinct points, the discriminant (b^2 - 4ac) must be greater than zero. In this case, it is:

4k + 9 > 0

4k > -9

k > -9/4

Therefore, the correct answer from the given options is k > − 2. This means that for the line and the parabola to intersect at two distinct points, the value of k must be greater than -2.

In summary, as a scientist, I can explain that the intersection points and finding the unknown variable in this scenario involve solving a quadratic equation and using the discriminant to determine the conditions for the two curves to intersect at two distinct points. The value of k must be greater than -2 for the two curves to intersect at two distinct points.