Q1. What are the solutions to the equation x^2 - 6x + 9 = 0?
Solution:
The equation can be factored as (x - 3)(x - 3) = 0. Thus, the solution is x = 3 with multiplicity 2.
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Q2. What is the value of x in the equation 3x + 4 = 10?
Solution:
To solve 3x + 4 = 10, subtract 4 from both sides: 3x = 6. Then divide by 3: x = 2.
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Q3. Factor the polynomial x^2 + 7x + 10.
Solution:
To factor, we look for two numbers that multiply to 10 and add to 7. The numbers are 5 and 2. Thus, the factorization is (x + 5)(x + 2).
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Q4. What are the roots of the equation 2x^2 + 3x - 2 = 0?
Solution:
Using the quadratic formula x = (-b ± √(b^2 - 4ac)) / 2a, where a = 2, b = 3, c = -2, we find the roots to be x = 2 and x = -1.5.
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Q5. What are the roots of the equation x^2 + 4x + 4 = 0?
Solution:
The equation can be factored as (x + 2)(x + 2) = 0. Thus, the root is x = -2 with multiplicity 2.
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Q6. What is the vertex of the quadratic function y = 2x^2 - 8x + 5?
Solution:
The vertex can be found using the formula x = -b/(2a). Here, a = 2 and b = -8, so x = 8/4 = 2. Plugging x = 2 into the equation gives y = 2(2)^2 - 8(2) + 5 = -3. Thus, the vertex is (2, -3).
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Q7. What are the roots of the quadratic equation 2x^2 - 8x = 0?
Solution:
Factoring gives 2x(x - 4) = 0. Thus, the roots are x = 0 and x = 4.
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Q8. Which of the following is a factor of the polynomial x^2 - 4?
Solution:
The polynomial x^2 - 4 can be factored as (x - 2)(x + 2). Therefore, x - 2 is a factor.
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Q9. What are the roots of the equation x^2 + 6x + 9 = 0?
Solution:
This is a perfect square: (x + 3)(x + 3) = 0. Thus, the root is x = -3 (with multiplicity 2).
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Q10. What is the solution to the equation 2x^2 - 8 = 0?
Solution:
First, add 8 to both sides: 2x^2 = 8. Then divide by 2: x^2 = 4. Taking the square root gives x = ±4.
