Let's solve the equation [tex]\( x^2 + 6x = 81 \)[/tex] by completing the square. Here's a step-by-step walkthrough:
### Step 1: Move the constant term to the left side
First, let's rewrite the equation so that we have all the terms involving [tex]\( x \)[/tex] on one side and the constants on the other side:
[tex]\[ x^2 + 6x - 81 = 0 \][/tex]
### Step 2: Bring the constant term to the right side
Rewrite the equation to isolate the terms involving [tex]\( x \)[/tex] on the left side:
[tex]\[ x^2 + 6x = 81 \][/tex]
### Step 3: Add and subtract the square of half the coefficient of [tex]\( x \)[/tex]
To complete the square, we need to add and subtract the square of half the coefficient of [tex]\( x \)[/tex]. The coefficient of [tex]\( x \)[/tex] is 6, half of it is 3, and its square is 9. So, add and subtract 9 on the left side:
[tex]\[ x^2 + 6x + 9 - 9 = 81 \][/tex]
[tex]\[ x^2 + 6x + 9 = 81 + 9 \][/tex]
### Step 4: Factor the left-hand side
The left-hand side of the equation, [tex]\( x^2 + 6x + 9 \)[/tex], can be factored as a perfect square trinomial:
[tex]\[ (x + 3)^2 = 90 \][/tex]
### Step 5: Solve for [tex]\( x \)[/tex]
Now, we have the equation in the form of a perfect square:
[tex]\[ (x + 3)^2 = 90 \][/tex]
To find the values of [tex]\( x \)[/tex], take the square root of both sides:
[tex]\[ x + 3 = \pm \sqrt{90} \][/tex]
Thus, we get two solutions:
[tex]\[ x + 3 = \sqrt{90} \quad \text{and} \quad x + 3 = -\sqrt{90} \][/tex]
Simplify the square root of 90:
[tex]\[ \sqrt{90} = 3\sqrt{10} \][/tex]
So, the solutions are:
[tex]\[ x + 3 = 3\sqrt{10} \quad \Rightarrow \quad x = 3\sqrt{10} - 3 \][/tex]
[tex]\[ x + 3 = -3\sqrt{10} \quad \Rightarrow \quad x = -3\sqrt{10} - 3 \][/tex]
### Final Solutions
Therefore, the solutions to the equation [tex]\( x^2 + 6x = 81 \)[/tex] by completing the square are:
[tex]\[ x = 3\sqrt{10} - 3 \quad \text{and} \quad x = -3\sqrt{10} - 3 \][/tex]
