An Exercise on Quadratic Equations | Properties of Roots (Sum & Product, Discriminant)

Section A(2) & Section B Practice

Question 1 MC
If \( \alpha \) and \( \beta \) are roots of \( x^2 + 8x - 3 = 0 \), find \( \alpha + \beta \).
A. \( -8 \)     B. \( -3 \)     C. \( 3 \)     D. \( 8 \)

Question 2 MC
If \( \alpha \) and \( \beta \) are roots of \( 4x^2 - 3x + 2 = 0 \), find \( \alpha\beta \).
A. \( \frac{1}{2} \)     B. \( -\frac{3}{4} \)     C. \( \frac{3}{4} \)     D. \( -\frac{1}{2} \)

Question 3 Short Answer
Construct a quadratic equation with roots \( 7 \) and \( -4 \).

Question 4 Short Answer
If \( \alpha \) and \( \beta \) are roots of \( x^2 - 5x + 3 = 0 \), find \( \frac{1}{\alpha} + \frac{1}{\beta} \).

Question 5 MC
The roots of \( x^2 + kx + 10 = 0 \) are \( 2 \) and \( 5 \). Find \( k \).
A. \( -7 \)     B. \( -3 \)     C. \( 3 \)     D. \( 7 \)

Question 6 Short Answer
If \( \alpha \) and \( \beta \) are roots of \( 2x^2 - 7x + 5 = 0 \), find \( \alpha^2 + \beta^2 \).

Question 7 MC
Which statement about the roots of \( x^2 + x - 6 = 0 \) is true?
A. Both roots are positive     B. Both roots are negative     C. One positive, one negative     D. No real roots

Question 8 Short Answer
If one root of \( x^2 - 10x + k = 0 \) is 3, find the other root and the value of \( k \).

Question 9 Short Answer
Determine the signs of the roots of \( x^2 - 2x - 8 = 0 \).

Question 10 Short Answer
If \( \alpha \) and \( \beta \) are roots of \( x^2 - 4x + 1 = 0 \), find \( \alpha^3 + \beta^3 \).