Complete the square to solve the following quadratic equation: x^2−6x−20=20

Question

Complete the square to solve the following quadratic equation:  x^2−6x−20=20
 .(1 point)
Responses

x=52
, x=−46
x equals 52 ,  x equals negative 46

x=3+210−−√, 3−210−−√
x equals 3 plus 2 Start Root 10 End Root comma 3 minus 2 Start Root 10 End Root

x=−10
, x=4
x equals negative 10 ,  x equals 4

x=10
, x=−4
x equals 10 ,  x equals negative 4

GPT-4o mini · Answer

To solve the quadratic equation $ x^2 - 6x - 20 = 20 $ by completing the square, follow these steps: 1. First, move all terms to one side of the equation: $$ x^2 - 6x - 20 - 20 = 0 $$ This simplifies to: $$ x^2 - 6x - 40 = 0 $$ 2. Next, we isolate the $ x^2 $ and $ x $ terms: $$ x^2 - 6x = 40 $$ 3. Now, we will complete the square. Take half of the coefficient of $ x $ (which is $-6$), square it, and add it to both sides. Half of $-6$ is $-3$, and squaring it gives $9$: $$ x^2 - 6x + 9 = 40 + 9 $$ This simplifies to: $$ (x - 3)^2 = 49 $$ 4. Now solve for $ x $ by taking the square root of both sides: $$ x - 3 = \pm 7 $$ 5. This gives us two equations: - $ x - 3 = 7 $ which leads to $ x = 10 $ - $ x - 3 = -7 $ which leads to $ x = -4 $ So, the solutions to the equation are: $$ x = 10 \quad \text{and} \quad x = -4 $$ The correct response is: **x = 10, x = -4**.