Alright, let's go through the steps to complete the square and solve the equation [tex]\(x^2 + 14x = -13\)[/tex].
### Step 1: Write down the given equation:
[tex]\[ x^2 + 14x = -13 \][/tex]
### Step 2: Move the constant term to the right side of the equation to set up for completing the square:
[tex]\[ x^2 + 14x + \_\_ = -13 + \_\_ \][/tex]
### Step 3: Find the value that completes the square.
To complete the square, we need to take the coefficient of [tex]\(x\)[/tex] (which is 14), divide it by 2, and then square it:
[tex]\[ \left(\frac{14}{2}\right)^2 = 7^2 = 49 \][/tex]
### Step 4: Add and subtract this value inside the equation:
[tex]\[ x^2 + 14x + 49 = -13 + 49 \][/tex]
### Step 5: Rewrite the left side as a perfect square and simplify the right side:
[tex]\[ (x + 7)^2 = 36 \][/tex]
### Step 6: Solve for [tex]\(x\)[/tex]:
To solve [tex]\((x + 7)^2 = 36\)[/tex], we take the square root of both sides. Remember, we will have two solutions because we take both the positive and negative square roots:
[tex]\[ x + 7 = \pm \sqrt{36} \][/tex]
[tex]\[ x + 7 = \pm 6 \][/tex]
Thus, we get two equations to solve:
[tex]\[ x + 7 = 6 \][/tex]
[tex]\[ x + 7 = -6 \][/tex]
Solving these:
[tex]\[ x = 6 - 7 \Rightarrow x = -1 \][/tex]
[tex]\[ x = -6 - 7 \Rightarrow x = -13 \][/tex]
### Summary:
The given equation [tex]\[x^2 + 14x = -13\][/tex] is equivalent to [tex]\((x + 7)^2 = 36\)[/tex].
The values of [tex]\(x\)[/tex] that make this equation true are [tex]\( -1 \)[/tex] and [tex]\(-13\)[/tex].
So, filling in the blanks:
The given equation is equivalent to [tex]\( (x + 7)^2 = 36 \)[/tex].
The values of [tex]\(x\)[/tex] that make this equation true are [tex]\( -1 \)[/tex] and [tex]\(-13\)[/tex].
