Sure, let's solve the equation [tex]\( x^2 - 16x + 60 = -12 \)[/tex] by completing the square step-by-step.
### Step 1: Subtract 60 from each side
First, let's isolate the quadratic term by moving the constant term on the left side to the other side:
[tex]\[ x^2 - 16x + 60 - 60 = -12 - 60 \][/tex]
Simplify:
[tex]\[ x^2 - 16x = -72 \][/tex]
### Step 2: Complete the square
To complete the square, we need to add and subtract a specific value on the left-hand side to make it a perfect square trinomial. The value to add and subtract is [tex]\(\left(\frac{b}{2}\right)^2\)[/tex], where [tex]\(b\)[/tex] is the coefficient of [tex]\(x\)[/tex]. Here, [tex]\(b = -16\)[/tex], so:
[tex]\[ \left(\frac{-16}{2}\right)^2 = 64 \][/tex]
Add and subtract 64 from both sides:
[tex]\[ x^2 - 16x + 64 - 64 = -72 + 64 \][/tex]
This simplifies to:
[tex]\[ x^2 - 16x + 64 = -8 \][/tex]
### Step 3: Write the left side as a squared term
Now we can rewrite the left-hand side as a perfect square trinomial:
[tex]\[ (x - 8)^2 = -8 \][/tex]
### Step 4: Take the square root of both sides
To solve [tex]\( (x - 8)^2 = -8 \)[/tex], take the square root of both sides:
[tex]\[ x - 8 = \pm \sqrt{-8} \][/tex]
Recall that [tex]\(\sqrt{-8} = \sqrt{8} \cdot i\)[/tex], where [tex]\(i\)[/tex] is the imaginary unit:
[tex]\[ x - 8 = \pm \sqrt{8} \cdot i \][/tex]
So,
[tex]\[ x = 8 \pm \sqrt{8} \cdot i \][/tex]
### Solutions
The solutions to the equation are:
[tex]\[ x_1 = 8 + \sqrt{8} \cdot i \][/tex]
[tex]\[ x_2 = 8 - \sqrt{8} \cdot i \][/tex]
Numerically,
[tex]\(\sqrt{8} \approx 2.8284271247461903\)[/tex], so:
[tex]\[ x_1 \approx 8 + 2.8284271247461903i \][/tex]
[tex]\[ x_2 \approx 8 - 2.8284271247461903i \][/tex]
Thus, the solutions are approximately:
[tex]\[ x_1 \approx (8 + 2.8284271247461903i) \][/tex]
[tex]\[ x_2 \approx (8 - 2.8284271247461903i) \][/tex]
