To convert the quadratic equation [tex]\( y = 8x^2 + 32x + 17 \)[/tex] to its vertex form, we need to complete the square. Let's go through this process step-by-step.
Step 1: Factor out the leading coefficient from the [tex]\( x^2 \)[/tex] and [tex]\( x \)[/tex] terms.
[tex]\[ y = 8(x^2 + 4x) + 17 \][/tex]
Step 2: Form a perfect-square trinomial inside the parentheses. To do this, we need to complete the square on [tex]\( x^2 + 4x \)[/tex].
The term inside the parentheses is [tex]\( x^2 + 4x \)[/tex].
To complete the square, take half of the linear coefficient (4), square it, and add it inside the parentheses. So, [tex]\((\frac{4}{2})^2 = 4.0\)[/tex].
Step 3: Add and subtract this squared term inside the parentheses:
[tex]\[ y = 8(x^2 + 4x + 4.0 - 4.0) + 17 \][/tex]
Step 4: Rearrange the equation to isolate the perfect-square trinomial and factor it:
[tex]\[ y = 8((x^2 + 4x + 4.0) - 4.0) + 17 \][/tex]
Step 5: Distribute the 8 through the trinomial and simplify:
[tex]\[ y = 8(x^2 + 4x + 4.0) - 8(4.0) + 17 \][/tex]
Step 6: Simplify the constants:
[tex]\[ y = 8(x^2 + 4x + 4.0) - 32.0 + 17 \][/tex]
Therefore, the complete statements are:
[tex]\[ y = 8(x^2 + 4x + 4.0) + 17 - 32.0 \][/tex]
[tex]\[ y = 8(x^2 + 4x + 4.0) - 15 \][/tex]
And after completing the square, the final vertex form of the quadratic equation is:
[tex]\[ y = 8(x + 2)^2 - 15 \][/tex]
