To express the quadratic expression [tex]\(x^2 - 8x + 10\)[/tex] in the form [tex]\((x + a)^2 + b\)[/tex], follow these steps:
### Step 1: Write the Quadratic Expression
Starting with the quadratic expression:
[tex]\[ x^2 - 8x + 10 \][/tex]
### Step 2: Identify the Coefficient of [tex]\(x\)[/tex]
Look at the coefficient of the linear term [tex]\(x\)[/tex], which is [tex]\(-8\)[/tex].
### Step 3: Complete the Square
To complete the square, take half of the coefficient of [tex]\(x\)[/tex] and square it. The coefficient of [tex]\(x\)[/tex] is [tex]\(-8\)[/tex], so:
[tex]\[ \left(\frac{-8}{2}\right)^2 = 16 \][/tex]
Now, add and subtract this square inside the expression:
[tex]\[ x^2 - 8x + 16 - 16 + 10 \][/tex]
### Step 4: Form a Perfect Square
Rewrite the expression so that it includes a perfect square trinomial:
[tex]\[ (x - 4)^2 - 16 + 10 \][/tex]
### Step 5: Simplify the Expression
Combine the constant terms outside the perfect square:
[tex]\[ (x - 4)^2 - 6 \][/tex]
### Step 6: Rewrite in the Desired Form
The quadratic expression [tex]\(x^2 - 8x + 10\)[/tex] is now written in the form [tex]\((x + a)^2 + b\)[/tex]. Here, [tex]\(a = -4\)[/tex] and [tex]\(b = -6\)[/tex]. Therefore:
[tex]\[ x^2 - 8x + 10 = (x - 4)^2 - 6 \][/tex]
Therefore, the expression [tex]\( x^2 - 8x + 10 \)[/tex] can be rewritten as:
[tex]\[ \boxed{(x - 4)^2 - 6} \][/tex]
This is the correctly completed square form of the given quadratic expression.
