Q1. If one root of the equation x^2 + px + 6 = 0 is 2, what is the value of p?
Solution:
If one root is 2, then the other root can be found using the product of the roots: 2 * r = 6, so r = 3. The sum of the roots is 2 + 3 = -p, thus p = -5.
⏱ Time: 0s
Q2. Solve for x: x^2 + 2x - 8 = 0.
Solution:
Factoring gives (x + 4)(x - 2) = 0. Thus, the solutions are x = -4 and x = 2.
⏱ Time: 0s
Q3. What is the factored form of the quadratic expression x^2 - 7x + 10?
Solution:
To factor, we look for two numbers that multiply to 10 and add to -7. The numbers -2 and -5 work, so the factored form is (x - 2)(x - 5).
⏱ Time: 0s
Q4. Solve the equation x^2 + 4x + 4 = 0.
Solution:
This can be factored as (x + 2)(x + 2) = 0. Therefore, x = -2 is a double root.
⏱ Time: 0s
Q5. What is the sum of the roots of the quadratic equation x^2 + 3x - 10 = 0?
Solution:
The sum of the roots of a quadratic equation ax^2 + bx + c = 0 is given by -b/a. Here, b = 3 and a = 1, so the sum is -3/1 = -3.
⏱ Time: 0s
Q6. If one root of the equation x^2 + px + q = 0 is 3, what is the value of p if the other root is 1?
Solution:
Using the sum of the roots, p = -(3 + 1) = -4. The product of the roots gives q = 3 * 1 = 3.
⏱ Time: 0s
Q7. What is the sum of the roots of the equation x^2 - 8x + 15 = 0?
Solution:
The sum of the roots can be found using -b/a. Here, b = -8 and a = 1. Thus, the sum is 8.
⏱ Time: 0s
Q8. What is the discriminant of the quadratic equation 3x^2 + 6x + 2 = 0?
Solution:
The discriminant is given by b^2 - 4ac. Here, a = 3, b = 6, c = 2. Thus, the discriminant = 6^2 - 4*3*2 = 36 - 24 = 12.
⏱ Time: 0s
Q9. What are the solutions to the equation x^2 - 10x + 25 = 0?
Solution:
The equation can be factored as (x - 5)(x - 5) = 0. Thus, the solution is x = 5 (a repeated root).
⏱ Time: 0s
Q10. What is the factored form of the quadratic equation x^2 + 6x + 9?
Solution:
This quadratic can be factored as (x + 3)(x + 3) or (x + 3)^2.
