Quadratic Formula Practice Problems

What this problem type covers

Every problem here is a quadratic equation ax² + bx + c = 0 to be solved with the quadratic formula. Difficulty rises from integer roots with a leading coefficient of 1, through fractional roots, to irrational roots where the discriminant is not a perfect square.

Example problems

One example from each difficulty band, with worked solutions.

These problems are generated automatically. The answers are derived as each problem is built and are checked by tests, but mistakes are still possible — please tell us if you find one.

1. 1 · ★1
   Solve for x using the quadratic formula.
   x² − 5x + 6 = 0
   • ①x = 3
   • ②x = −5
   • ③x = 3 or x = 2
   • ④x = 5
   • ⑤x = 1
   Show solution

   ③x = 3 or x = 2

   1. Identify the coefficients: a = 1, b = −5, c = 6
   2. Find the discriminant: D = b² − 4ac = (−5)² − 4·1·(6) = 1
   3. Substitute into x = (−b ± √D) / 2a: x = (5 ± √1) / 2
   4. Simplify: x = 3 or x = 2
2. 2 · ★2
   Solve for x using the quadratic formula.
   x² − 10x + 16 = 0
   • ①x = 36
   • ②x = 8 or x = 2
   • ③x = 10
   • ④x = −10
   • ⑤x = 8
   Show solution

   ②x = 8 or x = 2

   1. Identify the coefficients: a = 1, b = −10, c = 16
   2. Find the discriminant: D = b² − 4ac = (−10)² − 4·1·(16) = 36
   3. Substitute into x = (−b ± √D) / 2a: x = (10 ± √36) / 2
   4. Simplify: x = 8 or x = 2
3. 3 · ★3
   Solve for x using the quadratic formula.
   3x² − 192 = 0
   • ①x = −192
   • ②x = −8 or x = 8
   • ③x = 2,304
   • ④x = 0
   • ⑤x = −8
   Show solution

   ②x = −8 or x = 8

   1. Identify the coefficients: a = 3, b = 0, c = −192
   2. Find the discriminant: D = b² − 4ac = (0)² − 4·3·(−192) = 2304
   3. Substitute into x = (−b ± √D) / 2a: x = (0 ± √2304) / 6
   4. Simplify: x = −8 or x = 8
4. 4 · ★4
   Solve for x using the quadratic formula.
   6x² + 10x + 4 = 0
   • ①x = −10
   • ②x = −1 or x = −2/3
   • ③x = −1
   • ④x = 4
   • ⑤x = 10
   Show solution

   ②x = −1 or x = −2/3

   1. Identify the coefficients: a = 6, b = 10, c = 4
   2. Find the discriminant: D = b² − 4ac = (10)² − 4·6·(4) = 4
   3. Substitute into x = (−b ± √D) / 2a: x = (−10 ± √4) / 12
   4. Simplify: x = −1 or x = −2/3
5. 5 · ★5
   Solve for x using the quadratic formula.
   2x² + 6x − 5 = 0
   • ①x = (3 ± √19) / 2
   • ②x = (−3 ± √19) / 2
   • ③x = −3 ± √19
   • ④x = −6
   • ⑤x = 76
   Show solution

   ②x = (−3 ± √19) / 2

   1. Identify the coefficients: a = 2, b = 6, c = −5
   2. Find the discriminant: D = b² − 4ac = (6)² − 4·2·(−5) = 76
   3. Substitute into x = (−b ± √D) / 2a: x = (−6 ± √76) / 4
   4. Simplify: x = (−3 ± √19) / 2

Where people usually go wrong

• Dropping the sign of b. The formula uses −b, so a negative b becomes positive — this is the single most common slip.
• Forgetting that the denominator is 2a, not 2. When the leading coefficient is not 1, this silently halves or doubles the answer.
• Squaring a negative b incorrectly. (−5)² is 25, not −25.
• Stopping at the discriminant. A positive non-square discriminant still needs the radical simplified, not left as √72.
• Leaving fractional roots unreduced. 6/4 and 3/2 are the same number, but only one of them is a finished answer.