Simultaneous Equations with One Quadratic · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Simultaneous equations with one quadratic

Mathematics · Algebra · ages 15-16
Name ______________________   Date ____________

  1. At an intersection with x = 3, the equations are y = x² - 2 and y = x + 4. Which calculation finds y with less arithmetic here?

    • Use 3 + 4 from the linear rule
    • Use 3² - 2 from the quadratic rule
    • Use neither rule because y is not needed
    • Use either rule only when x is negative

  2. A line is substituted into a curve and the result is x squared - 4x + 4 = 0. What is the discriminant of that equation?

    Answer: ______________

  3. If the substituted quadratic has a negative discriminant, the line and the curve do not meet.

    Circle one:   True   False

  4. Solve y = x squared - 8 and y = 2x. When x = 4, what is the y coordinate of that point?

    Answer: ______________

  5. Solve y = x squared - 5 and y = x + 1. Give the two points where the line meets the curve.

    • (3, 4) and (-2, 1)
    • (3, 4) and (2, -1)
    • (-3, 4) and (-2, -1)
    • (3, 4) and (-2, -1)

  6. A line and the curve y = x squared - 3 meet at the points (1, -2) and (3, 6). What is the equation of the line?

    • y = 4x - 2
    • y = 6x - 8
    • y = 2x - 4
    • y = 4x - 6

  7. If a nonvertical line is tangent to a quadratic-function graph, substitution produces a quadratic with a repeated root.

    Circle one:   True   False

  8. Solve y = x squared + 1 and y = 3x - 1. What are the x values where they meet?

    • 1 and -2
    • -1 and 2
    • 1 and 2
    • 3 and 2

  9. Leo solves y = x squared - 1 and y = 3x - 3. He substitutes to get x squared - 3x + 2 = 0, finds x = 1 or x = 2, and writes the points as (1, 1) and (2, 3). Where is the slip?

    • The substituted equation is wrong
    • The x values are wrong
    • One of the y values is wrong
    • His points are both correct

  10. Solve y = 4 - x squared and y = x + 2. Give the points where the line meets the curve.

    • (1, 3) and (-2, 0)
    • (1, 3) and (-2, 2)
    • (1, 3) and (2, 0)
    • (-1, 3) and (-2, 0)
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Simultaneous equations with one quadratic W1-mt_D89ql2sJvc-s1

  1. Use 3 + 4 from the linear rule · Both rules give 7, but the linear rule needs only 3 + 4.
  2. 0 · b squared is 16 and 4ac is 16, so the discriminant is 16 - 16, which is 0.
  3. True · A negative discriminant means the quadratic has no real roots, so there are no x values where the line and curve agree.
  4. 8 · Put x = 4 into the linear equation: y = 2 times 4, which is 8.
  5. (3, 4) and (-2, -1) · The x values are 3 and -2, and y = x + 1 gives y = 4 and y = -1.
  6. y = 4x - 6 · The gradient is (6 - (-2)) / (3 - 1), which is 4, and the line through (1, -2) with gradient 4 is y = 4x - 6.
  7. True · In this setting the tangent contact corresponds to a repeated root of the substituted quadratic.
  8. 1 and 2 · Substituting gives x squared + 1 = 3x - 1. Collecting gives x squared - 3x + 2 = 0, which factorises to (x - 1)(x - 2).
  9. One of the y values is wrong · When x = 1, the linear equation gives y = 0, not 1, so the points should be (1, 0) and (2, 3).
  10. (1, 3) and (-2, 0) · Substituting gives x squared + x - 2 = 0, so x is 1 or -2, and y = x + 2 gives y = 3 and y = 0.
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