Question

Use the complete the square method to find the vertex of the following parabola: Note: There will be some extra steps becouse the coefficient of x2 is not 7 . y=2x2−8x+10 Step 1:Separate non-x-terms from x-terms Step 2: Divide off the coefficient of x2 y−10=2x2−8x There's a slight problem with completing the square at this point, and it's because we have a coefficient of x2 that's not 7 . Divide both sides of the equation by 2 so that we can proceed with completing the square.

Answer 1

The vertex form of the equation is y = 2(x - 2)^2 + 2, where the vertex is located at (2, 2).

To complete the square for the quadratic equation y = 2x^2 - 8x + 10, we can follow the steps provided:

Step 1: Separate non-x terms from x terms:

y - 10 = 2x^2 - 8x

Step 2: Divide off the coefficient of x^2:

Divide both sides of the equation by 2:

(1/2)(y - 10) = x^2 - 4x

Now we can proceed with completing the square. Let's continue:

Step 3: Take half of the coefficient of x, square it, and add it to both sides of the equation:

(1/2)(y - 10) + (4/2)^2 = x^2 - 4x + (4/2)^2

Simplifying the right side:

(1/2)(y - 10) + 4 = x^2 - 4x + 4

Step 4: Simplify the equation on the left side:

(1/2)(y - 10) + 4 = x^2 - 4x + 4

Distribute (1/2) on the left side:

(1/2)y - 5 + 4 = x^2 - 4x + 4

Combine like terms on the left side:

(1/2)y - 1 = x^2 - 4x + 4

Step 5: Rewrite the equation with the squared term factored:

x^2 - 4x + 4 = (x - 2)^2

Substituting this back into the equation:

(1/2)y - 1 = (x - 2)^2

Step 6: Rewrite the equation in vertex form:

(1/2)y = (x - 2)^2 + 1

Step 7: Multiply both sides by 2 to eliminate the fraction:

y = 2(x - 2)^2 + 2

The vertex form of the equation is y = 2(x - 2)^2 + 2, where the vertex is located at (2, 2).