Worksheet: Solving Cubic Equations Graphically

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In this worksheet, we will practice solving a cubic equation graphically.

Q1:

Use technology to plot the graph of 𝑓 ( 𝑥 ) = 𝑥 + 5 𝑥 − 1 0 0 3 2 , and use this graph to find the solutions to the equation 𝑥 + 5 𝑥 = 1 0 0 3 2 to two decimal places.

  • A 𝑥 = 7 . 0 3
  • B 𝑥 = − 1 2 . 8 1
  • C 𝑥 = − 7 . 0 3
  • D 𝑥 = 3 . 4 4
  • E 𝑥 = − 3 . 4 4

Q2:

Use technology to plot the graph of 𝑓 ( 𝑥 ) = 𝑥 + 3 𝑥 − 2 3 2 , and use this graph to find the solutions to the equation 𝑥 + 3 𝑥 = 2 3 2 to two decimal places.

  • A 𝑥 = − 2 . 0 0 , 𝑥 = 1 . 0 0 , 𝑥 = 0 . 0 0
  • B 𝑥 = − 0 . 7 3 , 𝑥 = 1 , 𝑥 = 2 . 7 3
  • C 𝑥 = − 3 . 5 6 , 𝑥 = 0 , 𝑥 = 0 . 5 6
  • D 𝑥 = − 2 . 7 3 , 𝑥 = − 1 , 𝑥 = 0 . 7 3
  • E 𝑥 = − 0 . 5 6 , 𝑥 = 0 , 𝑥 = 3 . 5 6

Q3:

The figure shows the graphs of the curve 𝑦 = 𝑥 ( 𝑥 − 1 ) ( 𝑥 + 2 ) and the line 𝑦 = 𝑘 ( 𝑥 − 1 ) for some 𝑘 < 0 .

The graphs intersect at three points, one of which has 𝑥 -coordinate 1. Find a quadratic equation whose roots are the 𝑥 -coordinates of the other two points of intersection.

  • A 𝑥 ( 𝑥 − 1 ) = 𝑘
  • B 𝑥 ( 𝑥 + 2 ) + 𝑘 = 0
  • C 𝑥 ( 𝑥 + 2 ) = 0
  • D 𝑥 ( 𝑥 + 2 ) − 𝑘 = 0
  • E 𝑥 ( 𝑥 + 2 ) − 𝑘 ( 𝑥 − 1 ) = 0

By considering the discriminant of this quadratic equation, find the slope of the line through ( 1 , 0 ) which is tangent to the curve at some other point.

At what point is this line tangent to the curve?

  • A ( 0 , 0 )
  • B ( 3 , 3 0 )
  • C ( − 1 , 2 )
  • D ( 2 , 8 )
  • E ( 1 , 0 )

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