Algebra 1 Practice Test: Completing the Square
- Completing the Square
- 10 questions
- 13 minutes
These 10 questions use completing the square to solve equations and rewrite quadratics in vertex form. Tip: take half of the x-coefficient, square it and add it to both sides.
Tap an option to mark your answer. You can change your mind, or tap the same option again to leave the question blank. The correct answers and explanations appear when you finish the test.Tap an option to answer; tap it again to leave the question blank.
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Question 1
What number should be added to x2 + 10x to make a perfect square trinomial?
Correct answer: D
Half of 10 is 5, and 52 = 25, so x2 + 10x + 25 = (x + 5)2. Adding 100 squares the whole coefficient instead of half of it.
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Question 2
What number completes the square for x2 − 7x?
Correct answer: A
Half of −7 is −72, and its square is 494. So x2 − 7x + 494 = (x − 72)2.
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Question 3
Solve x2 + 6x = 16 by completing the square.
Correct answer: B
Add 9 to both sides: (x + 3)2 = 25, so x + 3 = ±5 and x = 2 or x = −8. Check: 4 + 12 = 16 and 64 − 48 = 16.
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Question 4
Which is x2 + 8x + 3 written in vertex form?
Correct answer: C
x2 + 8x + 16 = (x + 4)2, so x2 + 8x + 3 = (x + 4)2 − 16 + 3 = (x + 4)2 − 13. Adding 16 instead of subtracting it gives the wrong constant 19.
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Question 5
What is the vertex of the parabola y = x2 − 4x + 1?
Correct answer: D
Complete the square: y = (x − 2)2 − 4 + 1 = (x − 2)2 − 3. The vertex of (x − h)2 + k is (h, k) = (2, −3); note the sign of h.
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Question 6
Solve x2 − 2x − 5 = 0.
Correct answer: A
Move 5: x2 − 2x = 5. Add 1: (x − 1)2 = 6, so x = 1 ± 6. Forgetting to add 1 to the right side gives 1 ± 5.
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Question 7
Dana completes the square to solve 2x2 + 12x − 14 = 0.
Which equation should she get?
Correct answer: C
Divide by 2 first: x2 + 6x − 7 = 0, so x2 + 6x = 7. Add 9: (x + 3)2 = 16. Moving −7 without changing its sign leads to the trap (x + 3)2 = 2.
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Question 8
After completing the square, x2 + 10x + 29 = 0 becomes (x + 5)2 = −4. What does this tell you?
Correct answer: D
A real number squared is never negative, so (x + 5)2 cannot equal −4. The answers −3 and −7 come from wrongly using +4.
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Question 9
x2 + bx + 81 is a perfect square trinomial. Which could be the value of b?
Correct answer: C
81 = 92, and (x + 9)2 = x2 + 18x + 81. The middle coefficient is twice 9, not 9 itself.
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Question 10
The height of a ball, in feet, after t seconds is h(t) = −t2 + 8t + 4.
What is the maximum height of the ball?
Correct answer: A
h(t) = −(t2 − 8t + 16) + 16 + 4 = −(t − 4)2 + 20. The maximum is 20 feet at t = 4 seconds; 4 feet is only the starting height.
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