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The Quadratic Formula — Easy Worksheet 1
Grade 9 · MathSheets Name: ____________________ Date: __________
Answer each question. Write two solutions like x = -1.5 or x = 4 and round to the nearest hundredth when needed.
How to solve
For ax2 + bx + c = 0, the solutions are x = (-b ± √(b2 - 4ac)) / (2a). The discriminant b2 - 4ac tells how many real solutions there are: positive means 2, zero means 1, negative means none.
Example
Find the discriminant of 2x² - 2x - 4 = 0.
Answer: 36
Not quite. The answer is
💡 Here a = 2, b = −3 and c = 2. The discriminant is b² − 4ac = (−3)² − 4(2)(2) = 9 − 16 = −7. So the discriminant is −7.
✓ Correct!
Not quite. The answer is
💡 Here a = 2, b = −3 and c = 2. The discriminant is b² − 4ac = (−3)² − 4(2)(2) = 9 − 16 = −7. So the discriminant is −7.
💡 How to solve: The Quadratic Formula
For ax2 + bx + c = 0, the solutions are x = (-b ± √(b2 - 4ac)) / (2a). The discriminant b2 - 4ac tells how many real solutions there are: positive means 2, zero means 1, negative means none.
Try a similar one
Find the discriminant of 3x² - x + 7 = 0.
✓ 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
💡 Here a = 9, b = 36 and c = 36. The discriminant is b² − 4ac = 36² − 4(9)(36) = 1,296 − 1,296 = 0. The discriminant is 0, so the ± part adds and subtracts 0 and both answers are the same: 1 real solution.
✓ Correct!
Not quite. The answer is
💡 Here a = 9, b = 36 and c = 36. The discriminant is b² − 4ac = 36² − 4(9)(36) = 1,296 − 1,296 = 0. The discriminant is 0, so the ± part adds and subtracts 0 and both answers are the same: 1 real solution.
💡 How to solve: The Quadratic Formula
For ax2 + bx + c = 0, the solutions are x = (-b ± √(b2 - 4ac)) / (2a). The discriminant b2 - 4ac tells how many real solutions there are: positive means 2, zero means 1, negative means none.
Try a similar one
Find the discriminant of 2x² + 2x + 11 = 0.
✓ 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
💡 Here a = 1, b = −8 and c = 15. The discriminant is b² − 4ac = (−8)² − 4(1)(15) = 64 − 60 = 4. Use the quadratic formula: x = (−(−8) ± √4) / (2 × 1) = 8 ± 22. The two answers are x = 8 − 22 = 6 ÷ 2 = 3 and x = 8 + 22 = 10 ÷ 2 = 5. So x = 3 or x = 5.
✓ Correct!
Not quite. The answer is
💡 Here a = 1, b = −8 and c = 15. The discriminant is b² − 4ac = (−8)² − 4(1)(15) = 64 − 60 = 4. Use the quadratic formula: x = (−(−8) ± √4) / (2 × 1) = 8 ± 22. The two answers are x = 8 − 22 = 6 ÷ 2 = 3 and x = 8 + 22 = 10 ÷ 2 = 5. So x = 3 or x = 5.
💡 How to solve: The Quadratic Formula
For ax2 + bx + c = 0, the solutions are x = (-b ± √(b2 - 4ac)) / (2a). The discriminant b2 - 4ac tells how many real solutions there are: positive means 2, zero means 1, negative means none.
Try a similar one
Find the discriminant of 2x² - 3x + 6 = 0.
✓ 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
💡 Here a = 1, b = −2 and c = −48. The discriminant is b² − 4ac = (−2)² − 4(1)(−48) = 4 + 192 = 196. Use the quadratic formula: x = (−(−2) ± √196) / (2 × 1) = 2 ± 142. The two answers are x = 2 − 142 = −12 ÷ 2 = −6 and x = 2 + 142 = 16 ÷ 2 = 8. So x = −6 or x = 8.
✓ Correct!
Not quite. The answer is
💡 Here a = 1, b = −2 and c = −48. The discriminant is b² − 4ac = (−2)² − 4(1)(−48) = 4 + 192 = 196. Use the quadratic formula: x = (−(−2) ± √196) / (2 × 1) = 2 ± 142. The two answers are x = 2 − 142 = −12 ÷ 2 = −6 and x = 2 + 142 = 16 ÷ 2 = 8. So x = −6 or x = 8.
💡 How to solve: The Quadratic Formula
For ax2 + bx + c = 0, the solutions are x = (-b ± √(b2 - 4ac)) / (2a). The discriminant b2 - 4ac tells how many real solutions there are: positive means 2, zero means 1, negative means none.
Try a similar one
Find the discriminant of x² - 5x + 7 = 0.
✓ 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
💡 Here a = 3, b = −1 and c = −8. The discriminant is b² − 4ac = (−1)² − 4(3)(−8) = 1 + 96 = 97. So the discriminant is 97.
✓ Correct!
Not quite. The answer is
💡 Here a = 3, b = −1 and c = −8. The discriminant is b² − 4ac = (−1)² − 4(3)(−8) = 1 + 96 = 97. So the discriminant is 97.
💡 How to solve: The Quadratic Formula
For ax2 + bx + c = 0, the solutions are x = (-b ± √(b2 - 4ac)) / (2a). The discriminant b2 - 4ac tells how many real solutions there are: positive means 2, zero means 1, negative means none.
Try a similar one
Find the discriminant of 3x² - 6x + 10 = 0.
✓ 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
💡 Here a = 2, b = −2 and c = 8. The discriminant is b² − 4ac = (−2)² − 4(2)(8) = 4 − 64 = −60. −60 < 0, and a negative number has no real square root: 0 real solutions.
✓ Correct!
Not quite. The answer is
💡 Here a = 2, b = −2 and c = 8. The discriminant is b² − 4ac = (−2)² − 4(2)(8) = 4 − 64 = −60. −60 < 0, and a negative number has no real square root: 0 real solutions.
💡 How to solve: The Quadratic Formula
For ax2 + bx + c = 0, the solutions are x = (-b ± √(b2 - 4ac)) / (2a). The discriminant b2 - 4ac tells how many real solutions there are: positive means 2, zero means 1, negative means none.
Try a similar one
Find the discriminant of 2x² - 5x + 1 = 0.
✓ 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
💡 Here a = 1, b = −1 and c = −42. The discriminant is b² − 4ac = (−1)² − 4(1)(−42) = 1 + 168 = 169. Use the quadratic formula: x = (−(−1) ± √169) / (2 × 1) = 1 ± 132. The two answers are x = 1 − 132 = −12 ÷ 2 = −6 and x = 1 + 132 = 14 ÷ 2 = 7. So x = −6 or x = 7.
✓ Correct!
Not quite. The answer is
💡 Here a = 1, b = −1 and c = −42. The discriminant is b² − 4ac = (−1)² − 4(1)(−42) = 1 + 168 = 169. Use the quadratic formula: x = (−(−1) ± √169) / (2 × 1) = 1 ± 132. The two answers are x = 1 − 132 = −12 ÷ 2 = −6 and x = 1 + 132 = 14 ÷ 2 = 7. So x = −6 or x = 7.
💡 How to solve: The Quadratic Formula
For ax2 + bx + c = 0, the solutions are x = (-b ± √(b2 - 4ac)) / (2a). The discriminant b2 - 4ac tells how many real solutions there are: positive means 2, zero means 1, negative means none.
Try a similar one
Find the discriminant of 2x² + 7x + 3 = 0.
✓ 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
💡 Here a = 1, b = −6 and c = −27. The discriminant is b² − 4ac = (−6)² − 4(1)(−27) = 36 + 108 = 144. Use the quadratic formula: x = (−(−6) ± √144) / (2 × 1) = 6 ± 122. The two answers are x = 6 − 122 = −6 ÷ 2 = −3 and x = 6 + 122 = 18 ÷ 2 = 9. So x = −3 or x = 9.
✓ Correct!
Not quite. The answer is
💡 Here a = 1, b = −6 and c = −27. The discriminant is b² − 4ac = (−6)² − 4(1)(−27) = 36 + 108 = 144. Use the quadratic formula: x = (−(−6) ± √144) / (2 × 1) = 6 ± 122. The two answers are x = 6 − 122 = −6 ÷ 2 = −3 and x = 6 + 122 = 18 ÷ 2 = 9. So x = −3 or x = 9.
💡 How to solve: The Quadratic Formula
For ax2 + bx + c = 0, the solutions are x = (-b ± √(b2 - 4ac)) / (2a). The discriminant b2 - 4ac tells how many real solutions there are: positive means 2, zero means 1, negative means none.
Try a similar one
Find the discriminant of 3x² + 11x - 8 = 0.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
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The Quadratic Formula — Easy Worksheet 1
Answer each question. Write two solutions like x = -1.5 or x = 4 and round to the nearest hundredth when needed.