Practice Questions
Concept Explanation
A quadratic equation has the form ax² + bx + c = 0 (a ≠ 0). Solutions are called roots.
Methods
- Factorization: Write as (px + q)(rx + s) = 0.
- Quadratic formula: x = (−b ± √(b² − 4ac)) / 2a
- Discriminant D = b² − 4ac: D > 0 two real roots; D = 0 one repeated root; D < 0 no real roots.
Example
x² − 5x + 6 = 0 ⇒ (x − 2)(x − 3) = 0 ⇒ x = 2 or x = 3.
Common Trap
Always set the equation to zero before factorizing. Sign errors in middle term split are frequent.
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Q1 EasySolve x² − 5x + 6 = 0.Explanation(x − 2)(x − 3) = 0 gives x = 2, 3.
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Q2 EasyWhat are the roots of the quadratic equation x² + 2x − 15 = 0?Explanation(x + 5)(x − 3) = 0 gives x = −5, 3.
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Q3 EasyWhat are the values of x that satisfy x² − 9 = 0?Explanationx² = 9, so x = ±3.
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Q4 EasyWhat is the discriminant of the quadratic equation x² + 4x + 4 = 0?ExplanationD = 4² − 4(1)(4) = 16 − 16 = 0.
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Q5 EasyWhat are the roots of the equation 2x² − 8x = 0?Explanation2x(x − 4) = 0 gives x = 0, 4.
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Q6 MediumHow many real roots does the quadratic equation x² + x + 1 = 0 have?ExplanationD = 1 − 4 = −3 < 0, so no real roots.
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Q7 MediumWhat are the roots of the quadratic equation x² − 6x + 9 = 0?Explanation(x − 3)² = 0 gives x = 3 (repeated root).
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Q8 MediumIf the quadratic equation x² − 7x + k = 0 has roots 3 and 4, what is the value of k?ExplanationProduct of roots = k = 3 × 4 = 12.
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Q9 MediumWhat are the roots of the quadratic equation x² − 4x − 5 = 0?Explanation(x − 5)(x + 1) = 0 gives x = 5, −1.
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Q10 MediumThe sum of the roots of the equation x² − 8x + 15 = 0 is:ExplanationSum = −(−8)/1 = 8.
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Q11 MediumWhat is the product of the roots of the quadratic equation x² + 3x − 10 = 0?ExplanationProduct = c/a = −10/1 = −10.
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Q12 HardWhat are the values of x that satisfy 3x² − 12 = 0?Explanationx² = 4, so x = ±2.
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Q13 HardIf one root of the quadratic equation x² − px + 12 = 0 is 3, what is the value of p?ExplanationOther root = 12/3 = 4. Sum p = 3 + 4 = 7.
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Q14 HardWhat is the discriminant of the quadratic equation x² + 2x + 5 = 0?ExplanationD = 4 − 20 = −16.
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DS1 Easy
What are the roots of the quadratic equation x² − 5x + 6 = 0?
Statement IOne root is 2.
Statement IIOne root is 3.
ExplanationNeed both roots 2 and 3 to confirm the solution set {2, 3}. -
DS2 Medium
How many real roots does the quadratic equation x² + bx + 9 = 0 have, where b is a constant?
Statement Ib = 6.
Statement IIb = 4.
ExplanationStatement I: D = 36 − 36 = 0 (one real root). Statement II: D = 16 − 36 < 0 (no real roots).
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DS3 Medium
What is the sum of the roots of the quadratic equation x² − px + 8 = 0, where p is a constant?
Statement Ip = 6.
Statement IIThe roots are 2 and 4.
ExplanationEach alone: p = 6 gives sum 6; roots 2 and 4 also give sum 6. -
DS4 Hard
What is the value of k if the quadratic equation x² − 5x + k = 0 has equal roots?
Statement IThe discriminant is zero.
Statement IIk = 25/4.
ExplanationEach alone determines k = 25/4 from D = 25 − 4k = 0.