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Completing the Square — Easy Worksheet 1
Grade 9 · MathSheets Name: ____________________ Date: __________
Answer each question. Write vertex form like (x + 3)^2 - 4 and two solutions like x = -7 or x = 1.
How to solve
For x2 + bx, add (b2)2 to make a perfect square: x2 + bx + (b2)2 = (x + b2)2. To solve x2 + bx + c = 0, move c to the right side, add (b2)2 to both sides, then take square roots of both sides.
Example
Find the value of c that makes x² + 6x + c a perfect-square trinomial.
Answer: 9
Not quite. The answer is
💡 Half of 6 is 3, and 3² = 9. So c = 9, and x² + 6x + 9 = (x + 3)².
✓ Correct!
Not quite. The answer is
💡 Half of 6 is 3, and 3² = 9. So c = 9, and x² + 6x + 9 = (x + 3)².
💡 How to solve: Completing the Square
For x2 + bx, add (b2)2 to make a perfect square: x2 + bx + (b2)2 = (x + b2)2. To solve x2 + bx + c = 0, move c to the right side, add (b2)2 to both sides, then take square roots of both sides.
Try a similar one
Find the value of c that makes x² - 4x + c a perfect-square trinomial.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 Half of −6 is −3, and (−3)² = 9. Add and subtract 9: x² − 6x − 2 = (x² − 6x + 9) − 9 − 2 = (x − 3)² − 11.
✓ Correct!
Not quite. The answer is
💡 Half of −6 is −3, and (−3)² = 9. Add and subtract 9: x² − 6x − 2 = (x² − 6x + 9) − 9 − 2 = (x − 3)² − 11.
💡 How to solve: Completing the Square
For x2 + bx, add (b2)2 to make a perfect square: x2 + bx + (b2)2 = (x + b2)2. To solve x2 + bx + c = 0, move c to the right side, add (b2)2 to both sides, then take square roots of both sides.
Try a similar one
Find the value of c that makes x² + 6x + c a perfect-square trinomial.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 Half of −10 is −5, and (−5)² = 25. So c = 25, and x² − 10x + 25 = (x − 5)².
✓ Correct!
Not quite. The answer is
💡 Half of −10 is −5, and (−5)² = 25. So c = 25, and x² − 10x + 25 = (x − 5)².
💡 How to solve: Completing the Square
For x2 + bx, add (b2)2 to make a perfect square: x2 + bx + (b2)2 = (x + b2)2. To solve x2 + bx + c = 0, move c to the right side, add (b2)2 to both sides, then take square roots of both sides.
Try a similar one
Find the value of c that makes x² - 6x + c a perfect-square trinomial.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 Add 12 to both sides: x² − 4x = 12. Half of −4 is −2, and (−2)² = 4. Add 4 to both sides: x² − 4x + 4 = 12 + 4 = 16, so (x − 2)² = 16. Take the square root of both sides: x − 2 = 4 or x − 2 = −4. Add 2 to both sides: x = 6 or x = −2. So x = −2 or x = 6.
✓ Correct!
Not quite. The answer is
💡 Add 12 to both sides: x² − 4x = 12. Half of −4 is −2, and (−2)² = 4. Add 4 to both sides: x² − 4x + 4 = 12 + 4 = 16, so (x − 2)² = 16. Take the square root of both sides: x − 2 = 4 or x − 2 = −4. Add 2 to both sides: x = 6 or x = −2. So x = −2 or x = 6.
💡 How to solve: Completing the Square
For x2 + bx, add (b2)2 to make a perfect square: x2 + bx + (b2)2 = (x + b2)2. To solve x2 + bx + c = 0, move c to the right side, add (b2)2 to both sides, then take square roots of both sides.
Try a similar one
Find the value of c that makes x² + 4x + c a perfect-square trinomial.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 Half of −4 is −2, and (−2)² = 4. So c = 4, and x² − 4x + 4 = (x − 2)².
✓ Correct!
Not quite. The answer is
💡 Half of −4 is −2, and (−2)² = 4. So c = 4, and x² − 4x + 4 = (x − 2)².
💡 How to solve: Completing the Square
For x2 + bx, add (b2)2 to make a perfect square: x2 + bx + (b2)2 = (x + b2)2. To solve x2 + bx + c = 0, move c to the right side, add (b2)2 to both sides, then take square roots of both sides.
Try a similar one
Find the value of c that makes x² + 8x + c a perfect-square trinomial.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 Half of 10 is 5, and 5² = 25. Add and subtract 25: x² + 10x + 26 = (x² + 10x + 25) − 25 + 26 = (x + 5)² + 1.
✓ Correct!
Not quite. The answer is
💡 Half of 10 is 5, and 5² = 25. Add and subtract 25: x² + 10x + 26 = (x² + 10x + 25) − 25 + 26 = (x + 5)² + 1.
💡 How to solve: Completing the Square
For x2 + bx, add (b2)2 to make a perfect square: x2 + bx + (b2)2 = (x + b2)2. To solve x2 + bx + c = 0, move c to the right side, add (b2)2 to both sides, then take square roots of both sides.
Try a similar one
Find the value of c that makes x² + 2x + c a perfect-square trinomial.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 Half of 8 is 4, and 4² = 16. So c = 16, and x² + 8x + 16 = (x + 4)².
✓ Correct!
Not quite. The answer is
💡 Half of 8 is 4, and 4² = 16. So c = 16, and x² + 8x + 16 = (x + 4)².
💡 How to solve: Completing the Square
For x2 + bx, add (b2)2 to make a perfect square: x2 + bx + (b2)2 = (x + b2)2. To solve x2 + bx + c = 0, move c to the right side, add (b2)2 to both sides, then take square roots of both sides.
Try a similar one
Find the value of c that makes x² + 6x + c a perfect-square trinomial.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 Add 9 to both sides: x² − 8x = 9. Half of −8 is −4, and (−4)² = 16. Add 16 to both sides: x² − 8x + 16 = 9 + 16 = 25, so (x − 4)² = 25. Take the square root of both sides: x − 4 = 5 or x − 4 = −5. Add 4 to both sides: x = 9 or x = −1. So x = −1 or x = 9.
✓ Correct!
Not quite. The answer is
💡 Add 9 to both sides: x² − 8x = 9. Half of −8 is −4, and (−4)² = 16. Add 16 to both sides: x² − 8x + 16 = 9 + 16 = 25, so (x − 4)² = 25. Take the square root of both sides: x − 4 = 5 or x − 4 = −5. Add 4 to both sides: x = 9 or x = −1. So x = −1 or x = 9.
💡 How to solve: Completing the Square
For x2 + bx, add (b2)2 to make a perfect square: x2 + bx + (b2)2 = (x + b2)2. To solve x2 + bx + c = 0, move c to the right side, add (b2)2 to both sides, then take square roots of both sides.
Try a similar one
Find the value of c that makes x² - 8x + c a perfect-square trinomial.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
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Completing the Square — Easy Worksheet 1
Answer each question. Write vertex form like (x + 3)^2 - 4 and two solutions like x = -7 or x = 1.