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3 Responses
Exactly-after factoring 2, adding 4 inside actually adds 2·4 = 8 overall, so you must subtract 8: 2(x^2 + 4x + 4) + 5 − 8 = 2(x + 2)^2 − 3. Quick nerdy check: expand back to 2x^2 + 8x + 8 − 3 = 2x^2 + 8x + 5, or note the vertex x = −b/(2a) = −2 gives value −3, matching 2(x + 2)^2 − 3.
You’re totally on the right track-this used to trip me up too because that sneaky 2 outside the brackets doubles whatever you add inside. Starting with 2x^2 + 8x + 5, factor to get 2(x^2 + 4x) + 5. To complete the square, add 4 inside because half of 4 is 2 and 2^2 = 4: 2(x^2 + 4x + 4) + 5. But adding 4 inside actually adds 2·4 = 8 to the whole expression, so yes-you should subtract 8 to balance. That gives 2(x + 2)^2 – 3. Quick checks: expand it back (2(x + 2)^2 – 3 = 2x^2 + 8x + 5), or use the vertex idea-since -b/(2a) = -8/4 = -2, the (x + 2) matches, and plugging x = -2 into the original gives 2·4 – 16 + 5 = -3, matching the constant. Want to try another one together, like 3x^2 – 12x + 1, and see which quick check you like better?
Yes-factor 2, add 4 inside to make (x+2)^2, and since that adds +8 overall you subtract 8 (some people think of it as “subtract 4 inside”), giving 2(x+2)^2 − 3; a quick check is to plug x=0 or expand to see you still get 5. Would you rather factor first like this, or divide by 2 to make the leading coefficient 1 before completing the square?