Use the completing the square method to transform the equation [x^2 + 2x = 13 into an equation of the form (x + 3)^2 = b where a and b are real numbers. What are the values of a and b?
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To transform the equation x^2 + 2x = 13 into the form (x + 3)^2 = b, we will complete the square by following these steps:
Step 1: Move the constant term to the other side of the equation.
x^2 + 2x - 13 = 0
Step 2: Take half of the coefficient of x, square it, and add it to both sides of the equation.
(x + 1)^2 - 1 - 13 = 0
Step 3: Simplify the equation.
(x + 1)^2 - 14 = 0
Step 4: Rearrange the equation to the form (x + 3)^2 = b by grouping the constant terms.
(x + 1)^2 = 14
Comparing (x + 1)^2 = 14 to (x + 3)^2 = b, we see that a = 1 and b = 14.
To transform the equation x^2 + 2x = 13 into the form (x + 3)^2 = b, we need to complete the square. Here are the steps:
1. Start with the equation: x^2 + 2x = 13.
2. Add a constant term on both sides of the equation to make completing the square possible. The constant term is the square of half the coefficient of x. In this case, the coefficient of x is 2, so we add (2/2)^2, which is 1.
The equation now becomes: x^2 + 2x + 1 = 13 + 1.
3. Simplify both sides of the equation: x^2 + 2x + 1 = 14.
4. Rewrite the left side of the equation as a perfect square trinomial. In this case, it is (x + 1)^2.
The equation becomes: (x + 1)^2 = 14.
5. Comparing it now with the desired form (x + 3)^2 = b, we see that a = 2 since the coefficient of x in (x + a)^2 is 2.
6. Thus, the values of a and b are: a = 2 and b = 14.
Therefore, we have transformed the equation x^2 + 2x = 13 into the equation (x + 3)^2 = 14 by completing the square.
x^2 + 2x = 13
x^2 + 2x + 1 = 13+1
(x+1)^2 = 14