Multiple Choice Questions
Quadratic Equations and Functions (Review)
Topic: Quadratic Equations and Functions (Review)
Grade: 10
Question 1:
What is the sum of the solutions of the quadratic equation 2x^2 + 5x – 3 = 0?
A) -5/2
B) -3/2
C) 3/2
D) 5/2
Answer: B) -3/2
Explanation: The sum of the solutions of a quadratic equation in the form ax^2 + bx + c = 0 can be found using the formula -b/a. In this case, the sum of the solutions is -5/2. Therefore, the correct answer is B) -3/2. For example, if we have the equation 2x^2 + 5x – 3 = 0, the solutions are x = 1/2 and x = -3. The sum of these solutions is 1/2 + (-3) = -3/2.
Question 2:
Which of the following quadratic functions has a vertex at (3, -4)?
A) f(x) = (x – 3)^2 – 4
B) f(x) = (x + 3)^2 – 4
C) f(x) = (x – 3)^2 + 4
D) f(x) = (x + 3)^2 + 4
Answer: A) f(x) = (x – 3)^2 – 4
Explanation: The vertex form of a quadratic function is f(x) = a(x – h)^2 + k, where (h, k) represents the vertex. In this case, the vertex is (3, -4), so the correct answer is A) f(x) = (x – 3)^2 – 4. For example, if we have the equation f(x) = (x – 3)^2 – 4, the vertex is (3, -4), and the parabola opens upwards.
Question 3:
Which of the following equations represents a quadratic function with no real solutions?
A) x^2 – 5x + 6 = 0
B) x^2 + 4x + 4 = 0
C) x^2 + 6x + 9 = 0
D) x^2 – 2x + 5 = 0
Answer: D) x^2 – 2x + 5 = 0
Explanation: A quadratic function has no real solutions if its discriminant is negative. The discriminant can be found using the formula b^2 – 4ac. In this case, the discriminant for equation D is -16, indicating that it has no real solutions. Therefore, the correct answer is D) x^2 – 2x + 5 = 0. For example, if we have the equation x^2 – 2x + 5 = 0, the discriminant is -16, and the solutions are complex numbers.
Question 4:
The graph of a quadratic function passes through the points (1, 3) and (4, 6). What is the equation of the function?
A) f(x) = x^2 + x + 2
B) f(x) = x^2 + 2x + 1
C) f(x) = 2x^2 + x + 1
D) f(x) = 2x^2 + 2x + 1
Answer: B) f(x) = x^2 + 2x + 1
Explanation: To find the equation of a quadratic function given two points, we can substitute the coordinates of the points into the general form of the quadratic function, ax^2 + bx + c, and solve for a, b, and c. In this case, substituting the points (1, 3) and (4, 6) gives us the equation system a + b + c = 3 and 16a + 4b + c = 6. Solving this system yields a = 1, b = 2, and c = 1, so the correct answer is B) f(x) = x^2 + 2x + 1. For example, if we have the equation f(x) = x^2 + 2x + 1, it passes through the points (1, 3) and (4, 6).
Question 5:
Which of the following quadratic functions has a maximum value?
A) f(x) = x^2 + 4x + 3
B) f(x) = -x^2 + 2x + 1
C) f(x) = 2x^2 – 3x + 1
D) f(x) = -2x^2 + x + 3
Answer: A) f(x) = x^2 + 4x + 3
Explanation: A quadratic function in the form ax^2 + bx + c has a maximum value if a < 0. In this case, the correct answer is A) f(x) = x^2 + 4x + 3. For example, if we have the equation f(x) = x^2 + 4x + 3, the coefficient of x^2 is 1, which is positive, indicating a parabola that opens upwards and has a minimum value.