Q1. What are the roots of the polynomial x^2 + 2x + 1?
Solution:
The polynomial can be factored as (x + 1)(x + 1) or (x + 1)^2. Thus, the roots are x = -1 and x = -1.
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Q2. Which of the following is the solution to the equation 3x + 4 = 10?
Solution:
Subtract 4 from both sides: 3x = 6. Then divide by 3: x = 2.
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Q3. What are the roots of the polynomial x^2 - 5x + 6?
Solution:
To find the roots, we can factor the polynomial: x^2 - 5x + 6 = (x - 2)(x - 3). Setting each factor to zero gives us x - 2 = 0 or x - 3 = 0, so the roots are x = 2 and x = 3.
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Q4. Which inequality represents the solution set for x^2 - 4 < 0?
Solution:
The inequality x^2 - 4 < 0 can be factored as (x - 2)(x + 2) < 0. The solution set is -2 < x < 2.
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Q5. Solve for x: 3x + 2 > 11.
Solution:
Subtract 2 from both sides: 3x > 9. Then divide by 3: x > 3.
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Q6. What is the sum of the roots of the polynomial x^2 + 6x + 8?
Solution:
The sum of the roots of a quadratic ax^2 + bx + c is given by -b/a. Here, b = 6 and a = 1, so the sum is -6/1 = -6.
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Q7. Which of the following is a factor of the polynomial x^2 + 4x + 4?
Solution:
The polynomial can be factored as (x + 2)(x + 2) or (x + 2)^2. Therefore, x + 2 is a factor.
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Q8. If x^2 - 4x + k has a double root, what is the value of k?
Solution:
For a double root, the discriminant must be zero: b^2 - 4ac = 0. Here, 4^2 - 4(1)(k) = 0. Thus, 16 - 4k = 0, leading to k = 4.
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Q9. Which of the following is the solution set for the inequality x + 2 > 3?
Solution:
To solve the inequality, subtract 2 from both sides: x > 1.
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Q10. If f(x) = x^2 - 4x + 4, what is f(2)?
Solution:
Substituting x = 2 into the function gives f(2) = 2^2 - 4(2) + 4 = 0.
