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. You found x = 3 as a solution of a line and a curve. Which equation is the safer choice for finding the matching y, and why?

    • The linear one, because it is simpler
    • The quadratic one, because it is the curve
    • Neither, y is not needed
    • Either, but only if x is positive

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

    Answer: ______________

  4. Solve y = x squared - 8 and y = 2x. Which equation do you get after substituting and collecting to one side?

    • x squared + 2x - 8 = 0
    • x squared - 2x + 8 = 0
    • x squared - 2x - 8 = 0
    • x squared - 8 = 0

  5. 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

  6. Maya solves y = x squared and y = 4x - 4. She substitutes to get x squared - 4x + 4 = 0 and concludes the line meets the curve at two points. Where is the slip?

    • x squared - 4x + 4 has a repeated root, not two
    • The substituted equation should be x squared + 4x + 4 = 0
    • She should have divided by 4 first
    • Her conclusion is correct

  7. Solve y = x squared - 2x - 3 and y = x + 1. What are the x values where they meet?

    • 4 and 1
    • 4 and -1
    • 1 and -3
    • 4 and -3

  8. The line y = x and the curve y = x squared + 1 never meet.

    Circle one:   True   False

  9. For which value of k does the line y = x + k touch the curve y = x squared?

    • 0.25
    • -1
    • 1
    • -0.25

  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)
LightMySky · lightmysky.comW1-mt_D89ql2sJvc-s1

Answer key

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

Simultaneous Equations with One Quadratic W1-mt_D89ql2sJvc-s1

  1. The linear one, because it is simpler · Both equations give the same y at the solution, but the linear one has less arithmetic, so it is the safer choice.
  2. 0 · b squared is 16 and 4ac is 16, so the discriminant is 16 - 16, which is 0.
  3. 8 · Put x = 4 into the linear equation: y = 2 times 4, which is 8.
  4. x squared - 2x - 8 = 0 · Substituting gives x squared - 8 = 2x, and collecting to one side gives x squared - 2x - 8 = 0.
  5. 1 and 2 · Substituting gives x squared + 1 = 3x - 1, so x squared - 3x + 2 = 0, which factorises to (x - 1)(x - 2).
  6. x squared - 4x + 4 has a repeated root, not two · x squared - 4x + 4 is (x - 2) squared, a repeated root, so the line touches the curve at one point, (2, 4).
  7. 4 and -1 · Substituting gives x squared - 2x - 3 = x + 1, so x squared - 3x - 4 = 0, which factorises to (x - 4)(x + 1), so x is 4 or -1.
  8. True · Substituting gives x squared - x + 1 = 0, whose discriminant is 1 - 4, which is -3, so there are no real meeting points.
  9. -0.25 · Substituting gives x squared - x - k = 0, and a tangent means the discriminant is zero: 1 + 4k = 0, so k is -0.25.
  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.
Worksheet · LightMySky