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Math 20-1

Math 20-1 · Worksheets

Mixed review · Quadratics and systems

Ten questions of mixed difficulty, covering Quadratic equations, Quadratic functions. Print it, or work through it on screen — the answer key starts on its own page.

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Mixed review · Quadratics and systems

Math 20-1 · maddyhelps.com

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Circle the best answer for each question. Show your work in the space provided.

  1. Factored, x² − 16 is

    1. a) (x² − 4)(x + 4)
    2. b) (x − 4)²
    3. c) (x − 4)(x + 4)
    4. d) (x − 8)(x + 2)

  2. The vertex of y = (x − 3)² + 5 is

    1. a) (5, 3)
    2. b) (3, 5)
    3. c) (−3, 5)
    4. d) (3, −5)

  3. The graph of y = −2(x + 1)² + 4

    1. a) opens downward and has a maximum value of 4
    2. b) opens upward and has a maximum value of 4
    3. c) opens upward and has a minimum value of −2
    4. d) opens downward and has a minimum value of 4

  4. The form that shows the vertex without any extra work is

    1. a) y = a(x − p)² + q
    2. b) y = ax² + bx + c
    3. c) y = a(x − r₁)(x − r₂)
    4. d) y = mx + b

  5. The graph of y = a(x − p)² + q has no x-intercepts when

    1. a) a > 0 and q < 0
    2. b) p = 0
    3. c) a > 0 and q > 0
    4. d) a < 0 and q > 0

  6. A parabola has vertex (−1, 4) and passes through (1, 0). Its equation is

    1. a) y = (x + 1)² + 4
    2. b) y = −(x − 1)² + 4
    3. c) y = −(x + 1)² − 4
    4. d) y = −(x + 1)² + 4

  7. For 2x² + 3x + 5 = 0, the discriminant shows the equation has

    1. a) two distinct real roots
    2. b) two equal rational roots
    3. c) exactly one real root
    4. d) no real roots

  8. A rectangular pen is built with 40 m of fencing. The greatest area it can enclose is

    1. a) 400 m²
    2. b) 100 m²
    3. c) 160 m²
    4. d) 80 m²

  9. The solution to x² − 5x + 6 = 0 is

    1. a) x = 5 or x = 6
    2. b) x = 2 or x = 3
    3. c) x = 1 or x = 6
    4. d) x = −2 or x = −3

  10. The vertex of y = 2x² − 8x + 3 is

    1. a) (2, −5)
    2. b) (4, 3)
    3. c) (2, 5)
    4. d) (−2, 27)

Answer key · Mixed review · Quadratics and systems

Math 20-1 · maddyhelps.com

  1. c) (x − 4)(x + 4) — This is a difference of squares, a² − b² = (a − b)(a + b), with a = x and b = 4. A difference of squares always splits; a sum of squares like x² + 16 does not factor over the real numbers.
  2. b) (3, 5) — In y = a(x − p)² + q the vertex is (p, q). The subtraction is built into the form, so x − 3 means p = 3, not −3. The sign flip catches people every time.
  3. a) opens downward and has a maximum value of 4 — The sign of a decides the direction: a = −2 is negative, so the parabola opens downward and the vertex is the highest point. The vertex is (−1, 4), so the maximum value of y is 4.
  4. a) y = a(x − p)² + q — Vertex form is built so that the vertex is (p, q). Standard form hides it — you have to complete the square or use x = −b/2a. Factored form shows the x-intercepts instead.
  5. c) a > 0 and q > 0 — An upward parabola whose lowest point is already above the x-axis never reaches it. The same is true the other way: a < 0 with q < 0. What matters is that a and q have the same sign.
  6. d) y = −(x + 1)² + 4 — Start from y = a(x + 1)² + 4 using the vertex, then substitute the point: 0 = a(2)² + 4, so 4a = −4 and a = −1. The negative a is expected, since the curve has to come back down to reach (1, 0).
  7. d) no real roots — b² − 4ac = 9 − 40 = −31. A negative discriminant means the parabola never crosses the x-axis, so there is no real solution. You can know this before attempting to solve, which saves time.
  8. b) 100 m² — With perimeter 40, the sides are x and 20 − x, so A = x(20 − x) = −x² + 20x. The vertex is at x = 10, giving A = 100 m². For a fixed perimeter the square always wins, which is worth remembering as a check.
  9. b) x = 2 or x = 3 — Two numbers multiply to 6 and add to −5: −2 and −3. So (x − 2)(x − 3) = 0 and the roots are 2 and 3. The roots come out with the opposite signs to the numbers in the brackets.
  10. a) (2, −5) — The axis of symmetry is x = −b/(2a) = 8/4 = 2. Substituting back: y = 2(4) − 16 + 3 = −5. So the vertex is (2, −5). Forgetting the minus sign in −b/(2a) sends you to x = −2.