Question

The steps for determining the center and radius of a circle using the completing the square method are shown below: Step 1 [original equation]: x2 − 10x + y2 + 12y = 20 Step 2 [group like terms]: (x2 − 10x) + (y2 + 12y) = 20 Step 3 [complete the quadratics]: (x2 − 10x + 25) + (y2 + 12y + 36) = 20 + (25 + 36) Step 4 [simplify the equation]: (x2 − 10x + 25) + (y2 + 12y + 36) = 64 Step 5 [factor each quadratic]: (x − 5)2 + (y + 6)2 = 82 Step 6 [identify the center and radius]: Center = (5, −6) Radius = 8 What is the first incorrect step, and how can it be fixed? Step 3, replace 25 with 5 and replace 36 with 6 Step 4, replace 64 with 81 Step 5, replace − 5 with + 5 and replace + 6 with − 6 All steps are correct.

Answer 1

We are given original equation: $x^2 -10x + y^2 + 12y = 20$

We need to find the enter and radius of a circle using the completing the square method.

The steps are as following :

Step 1 [original equation]: x^2 − 10x + y^2 + 12y = 20 .

Step 2 [group like terms]: (x^2 − 10x) + (y^2 + 12y) = 20

Step 3 [complete the quadratics]: (x^2 − 10x + 25) + (y^2 + 12y + 36) = 20 + (25 + 36).

Step 4 [simplify the equation]: (x^2 − 10x + 25) + (y^2 + 12y + 36) = 64.

Step 5 [factor each quadratic]: (x − 5)^2 + (y + 6)^2 = 8^2

Step 6 [identify the center and radius]: Center = (5, −6) Radius = 8.

Step 6 is incorrect.

The center should be (5,-6).

Replace − 5 with + 5 and replace + 6 with − 6.