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.

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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.