more than 4 weeks ago

1 view

1 view

Sort By

ChatGPT

more than 4 weeks ago

To find the coordinates where the parabola intersects the line y = 2x - 2, we need to set the equation of the parabola equal to the equation of the line and solve for x.

Let the equation of the parabola be y = ax^2 + bx + c. Since the parabola intersects the line, we have:

ax^2 + bx + c = 2x - 2

Now, we substitute y = 2x - 2 into the equation of the parabola:

a(2x - 2)^2 + b(2x - 2) + c = 2x - 2

Expanding and simplifying the equation, we get:

4ax^2 - 8ax + 4a + 2bx - 2b + c = 2x - 2

Now, we can equate the coefficients of x^2, x, and the constant term on both sides of the equation:

4a = 0 (coefficient of x^2)

2b - 8a = 2 (coefficient of x)

4a - 2b + c = -2 (constant term)

Solving these equations simultaneously, we get:

a = 0

b = 1

c = -2

Therefore, the equation of the parabola is y = x - 2.

To find the coordinates where the parabola intersects the line y = 2x - 2, we substitute y = 2x - 2 into the equation of the parabola:

2x - 2 = x - 2

Solving for x, we get:

x = 0

Now, substitute x = 0 back into the equation of the line to find the corresponding y-coordinate:

y = 2(0) - 2

y = -2

Therefore, the coordinates where the parabola intersects the line y = 2x - 2 are (0, -2).

Let the equation of the parabola be y = ax^2 + bx + c. Since the parabola intersects the line, we have:

ax^2 + bx + c = 2x - 2

Now, we substitute y = 2x - 2 into the equation of the parabola:

a(2x - 2)^2 + b(2x - 2) + c = 2x - 2

Expanding and simplifying the equation, we get:

4ax^2 - 8ax + 4a + 2bx - 2b + c = 2x - 2

Now, we can equate the coefficients of x^2, x, and the constant term on both sides of the equation:

4a = 0 (coefficient of x^2)

2b - 8a = 2 (coefficient of x)

4a - 2b + c = -2 (constant term)

Solving these equations simultaneously, we get:

a = 0

b = 1

c = -2

Therefore, the equation of the parabola is y = x - 2.

To find the coordinates where the parabola intersects the line y = 2x - 2, we substitute y = 2x - 2 into the equation of the parabola:

2x - 2 = x - 2

Solving for x, we get:

x = 0

Now, substitute x = 0 back into the equation of the line to find the corresponding y-coordinate:

y = 2(0) - 2

y = -2

Therefore, the coordinates where the parabola intersects the line y = 2x - 2 are (0, -2).

Google Gemini AI

more than 4 weeks ago

To find the coordinates where the parabola and the line intersect, we need to set their equations equal to each other and solve for x:

Parabola: y = x^2 + 1

Line: y = 2x - 2

Setting them equal:

x^2 + 1 = 2x - 2

x^2 - 2x + 3 = 0

This is a quadratic equation that can be solved using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic equation. In this case, a = 1, b = -2, and c = 3. Substituting these values into the formula, we get:

x = (-(-2) ± √((-2)^2 - 4(1)(3))) / 2(1)

x = (2 ± √(-8)) / 2

x = (2 ± 2i√2) / 2

x = 1 ± i√2

Substituting these values of x back into either the parabola equation or the line equation gives us the corresponding y-coordinates:

For x = 1 + i√2:

y = (1 + i√2)^2 + 1 = 2 + 2i√2

For x = 1 - i√2:

y = (1 - i√2)^2 + 1 = 2 - 2i√2

Therefore, the coordinates where the parabola and the line intersect are (1 + i√2, 2 + 2i√2) and (1 - i√2, 2 - 2i√2).

Parabola: y = x^2 + 1

Line: y = 2x - 2

Setting them equal:

x^2 + 1 = 2x - 2

x^2 - 2x + 3 = 0

This is a quadratic equation that can be solved using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic equation. In this case, a = 1, b = -2, and c = 3. Substituting these values into the formula, we get:

x = (-(-2) ± √((-2)^2 - 4(1)(3))) / 2(1)

x = (2 ± √(-8)) / 2

x = (2 ± 2i√2) / 2

x = 1 ± i√2

Substituting these values of x back into either the parabola equation or the line equation gives us the corresponding y-coordinates:

For x = 1 + i√2:

y = (1 + i√2)^2 + 1 = 2 + 2i√2

For x = 1 - i√2:

y = (1 - i√2)^2 + 1 = 2 - 2i√2

Therefore, the coordinates where the parabola and the line intersect are (1 + i√2, 2 + 2i√2) and (1 - i√2, 2 - 2i√2).