Question

Solve the following by completing the square or squaring both sides using steps illustrated in the lesson content.. Leave answers in radical (EXACT) form. Do not use decimals. -2x+6=-x^2​

Answer 1

Starting with the equation:

-2x + 6 = -x^2

First, we can rearrange it to put it in standard quadratic form:

x^2 - 2x - 6 = 0

To complete the square, we need to add and subtract a constant term that will make the left-hand side of the equation a perfect square. The constant we need to add is (b/2)^2, where b is the coefficient of x. In this case, b = -2, so:

x^2 - 2x + 1 - 1 - 6 = 0

The first three terms can be written as a perfect square:

(x - 1)^2 - 7 = 0

Add 7 to both sides:

(x - 1)^2 = 7

Now we can take the square root of both sides:

x - 1 = ±√7

Add 1 to both sides:

x = 1 ± √7

So the solutions to the equation are:

x = 1 + √7 or x = 1 - √7

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Answer 2

To solve the equation -2x + 6 = -x^2 by completing the square or squaring both sides, we need to rearrange the equation to get x^2 in one side and the other terms in the other side:

-x^2 + 2x - 6 = 0

Next, we need to complete the square by adding and subtracting (2/2)^2 = 1 from the left-hand side of the equation:

-(x^2 - 2x + 1 - 1) - 6 = 0

Now we can simplify the expression inside the parentheses:

-[(x - 1)^2 - 1] - 6 = 0

Distribute the negative sign:

-[(x - 1)^2 - 1] + 6 = 0

Simplify:

-(x - 1)^2 + 7 = 0

Next, we can square both sides to eliminate the square root:

(x - 1)^2 = 7

Finally, we can take the square root of both sides, remembering to include the plus or minus sign:

x - 1 = ±√7

Adding 1 to both sides:

x = 1 ± √7

Therefore, the solutions to the equation -2x + 6 = -x^2 are x = 1 + √7 and x = 1 - √7.