Practice Questions
Concept Explanation
For a quadratic ax² + bx + c = 0 with roots α and β, Vieta's formulas link roots to coefficients without solving fully.
Vieta's Formulas
- α + β = −b/a
- αβ = c/a
- x² − (sum)x + (product) = 0 for monic quadratics
Derived Results
- α² + β² = (α + β)² − 2αβ
- ( α − β )² = (α + β)² − 4αβ
Example
For x² − 5x + 6 = 0: sum = 5, product = 6, roots 2 and 3.
Common Trap
Sign matters: sum = −b/a, not b/a.
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Q1 Easy
For the quadratic equation x² − 7x + 10 = 0, what is the sum of the roots according to Vieta's formulas?
ExplanationSum = −(−7)/1 = 7. -
Q2 Easy
For the quadratic equation x² − 7x + 10 = 0, what is the product of the roots according to Vieta's formulas?
ExplanationProduct = 10/1 = 10. -
Q3 Medium
For the equation 2x² + 8x + 6 = 0, what is the sum of the roots?
ExplanationSum = −8/2 = −4. -
Q4 Medium
For the equation 2x² + 8x + 6 = 0, what is the product of the roots?
ExplanationProduct = 6/2 = 3. -
Q5 Easy
The quadratic equation x² + px + 12 = 0 has roots 3 and 4. What is the value of p?
ExplanationSum = −p = 7, so p = −7. -
Q6 Medium
A monic quadratic equation has roots whose sum is 9 and whose product is 14. Which equation represents this quadratic?
Explanationx² − (sum)x + product = x² − 9x + 14. -
Q7 Medium
One root of the equation x² − 6x + k = 0 is 2. What is the value of k?
ExplanationOther root = k/2. Sum 6 gives 2 + k/2 = 6, k = 8.
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Q8 Easy
If α and β are the roots of x² − 5x + 6 = 0, what is the value of α + β?
ExplanationSum = 5. -
Q9 Easy
If α and β are the roots of x² − 5x + 6 = 0, what is the value of αβ?
ExplanationProduct = 6. -
Q10 Medium
The roots of a quadratic satisfy α + β = 8 and αβ = 15. What is the value of α² + β²?
Explanation64 − 30 = 34. -
Q11 Hard
For what value of k does the equation x² − 2x + k = 0 have equal roots?
ExplanationD = 4 − 4k = 0 gives k = 1. -
Q12 Hard
The roots of a quadratic satisfy α + β = 10 and (α − β)² = 4. What is the value of αβ?
Explanation|α − β| = 2. (α + β)² − (α − β)² = 4αβ, so 100 − 4 = 4αβ, αβ = 24. -
Q13 Medium
For the quadratic equation x² − 5x + 1 = 0, what is the sum of the roots?
ExplanationSum = 5. -
Q14 Hard
One root of the equation x² − px + 20 = 0 is 4. What is the other root?
ExplanationProduct = 20, other root = 20/4 = 5.
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DS1 Easy
For the quadratic equation x² − 8x + 15 = 0, what is the sum of the roots?
Statement IThe coefficient of x is −8.
Statement IIThe roots are 3 and 5.
ExplanationEach alone gives sum 8. -
DS2 Medium
For the equation 3x² − 12x + 9 = 0, what is the product of the roots?
Statement Ia = 3 and c = 9.
Statement IIProduct = c/a.
ExplanationNeed both: product = 9/3 = 3.
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DS3 Medium
The equation x² + px + 12 = 0 has roots 3 and 4. What is the value of p?
Statement IOne root is 3.
Statement IIOne root is 4.
ExplanationNeed both roots to get sum 7 and p = −7. -
DS4 Hard
For what value of k does the equation x² − 6x + k = 0 have equal roots?
Statement IDiscriminant is zero.
Statement IIk = 9.
ExplanationEach alone: D = 36 − 4k = 0 gives k = 9.