• (3x - 2)(x + 1)

• (3x + 1)(x - 2)

• (3x + 2)(x - 1)

• (3x - 1)(x + 2)

Answer explanation

To factor the quadratic expression 3x^2 - 5x - 2, we look for two binomials. The correct factors are (3x + 1) and (x - 2), since their product gives the original expression.

• (0, 5)

• (5, 0)

• (-5, 0)

• (0, -5)

Answer explanation

The vertex of the quadratic equation in the form y = ax^2 + bx + c is given by the formula (-b/2a, f(-b/2a)). Here, a = -1 and b = 0, so the vertex is (0, 5). Thus, the correct answer is (0, 5).

• x = -2

• x = 2

• x = 0

• x = -1

Answer explanation

The axis of symmetry for a quadratic function in the form y = ax^2 + bx + c is given by x = -\frac{b}{2a}. Here, a = \frac{1}{2} and b = 2, so x = -\frac{2}{2 \cdot \frac{1}{2}} = -2. Thus, the correct answer is x = -2.

• f(x) = (x + 5)(x + 2)

• f(x) = (x + 3)(x + 7)

• f(x) = (x + 10)(x + 1)

• f(x) = (x + 1)(x + 21)

Answer explanation

To factor f(x) = x^2 + 10x + 21, we look for two numbers that multiply to 21 and add to 10. The numbers 3 and 7 fit this, so f(x) = (x + 3)(x + 7) reveals the zeros at x = -3 and x = -7.

Tags

CCSS.HSA-SSE.B.3B

CCSS.HSF-IF.C.8A

Answer explanation

To find the minimum value of f(x) = x^2 + 10x + 21, we complete the square: f(x) = (x + 5)^2 - 4. The vertex form reveals the minimum value at -4 when x = -5, confirming the correct choice.

Identify the characteristic of the quadratic function and its graph. The vertex is at:

• (-3, 3)

• (3, 3)

• (3, -3)

• (0, 0)

Answer explanation

The vertex of a quadratic function represents its maximum or minimum point. The correct choice, (-3, 3), indicates a vertex located at x = -3 and y = 3, which is a valid characteristic of a quadratic function.

• (3, -3)

• (-3, 3)

• (3, 3)

• (-3, -3)

Answer explanation

The vertex form of a quadratic equation is given by f(x) = a(x - h)^2 + k, where (h, k) is the vertex. Here, f(x) = (x - 3)^2 - 3, so the vertex is (3, -3). Thus, the correct answer is (3, -3).

• Length: x, width: x + 3

• Length: 2x, width: 3

Answer explanation

The area of a rectangle is given by length times width. The expression x^2 + 3 can be factored as x(x + 3), indicating that length = x and width = x + 3 is correct. Other options do not match the area expression.

Which of the following is equivalent to a quadratic function whose graph has a vertex (-2, -6) and passes through the point (1, -2)?

Answer explanation

The vertex form of a quadratic function is f(x) = a(x + 2)^2 - 6. Using the point (1, -2) to find 'a', we get -2 = a(1 + 2)^2 - 6, leading to a = \frac{4}{9}. Thus, f(x) = \frac{4}{9}(x + 2)^2 - 6.

Answer explanation

To convert the quadratic equation to vertex form, we complete the square. The expression simplifies to y = 5(x - 2)^2 + 6, confirming that the correct choice is y = 5(x - 2)^2 + 6.