Which of the following is equivalent to the expression above?
For quadratic “equivalent expression” questions, it is usually fastest to rewrite the given quadratic into vertex form by factoring out the leading coefficient from the and terms and then completing the square. Once you have it in the form , compare directly to the answer choices rather than expanding every option. Double-check your distribution of the outside coefficient and the final constant term, because small sign errors there are the most common traps on these problems.
Hints
Prepare to complete the square
Focus on the and terms. Can you factor out the coefficient of so that the quadratic inside the parentheses starts with ?
Work on the expression inside the parentheses
Once you have something like inside parentheses, think about how to rewrite it as a perfect square plus or minus a constant (completing the square).
Be careful with the outside coefficient
After completing the square inside the parentheses, remember that the -3 multiplies both terms inside. Distribute it and then combine any constant terms outside the parentheses.
Desmos Guide
Enter the original expression as a function
In Desmos, type f(x) = -3x^2 + 12x - 5 to graph the original quadratic.
Enter each answer choice as separate functions
Type g(x) = -3(x-2)^2 + 7, h(x) = -3(x+2)^2 + 7, p(x) = 3(x-2)^2 + 7, and q(x) = -3(x-2)^2 - 7. Each will appear as a different parabola.
Compare the graphs to find the match
Zoom out or adjust the view so you can clearly see all the parabolas. The correct choice is the one whose graph lies exactly on top of the graph of for all (same vertex, same direction of opening, same width). The others will be shifted, flipped, or vertically moved.
Step-by-step Explanation
Factor out the leading coefficient
Start with the expression:
Factor from the and terms:
Now you are set up to complete the square on inside the parentheses.
Complete the square inside the parentheses
Look at .
To complete the square:
- Take half of , which is .
- Square it: .
So:
Substitute this back into the expression:
Distribute -3 and combine constant terms
Distribute across the two terms inside the parentheses:
So the expression is now written as
This is almost in the same form as the answer choices; you just need to simplify the constants.
Simplify and match to an answer choice
Combine the constant terms and :
So the original expression is equivalent to
This matches choice A) , so A is the correct answer.