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Lesson: Solving Cubic Equations Graphically

Sample Question Videos

Worksheet • 3 Questions • 1 Video

Q1:

The figure shows the graphs of the curve 𝑦 = π‘₯ ( π‘₯ βˆ’ 1 ) ( π‘₯ + 2 ) and the line 𝑦 = π‘˜ ( π‘₯ βˆ’ 1 ) for some π‘˜ < 0 .

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

  • A π‘₯ ( π‘₯ + 2 ) βˆ’ π‘˜ = 0
  • B π‘₯ ( π‘₯ + 2 ) βˆ’ π‘˜ ( π‘₯ βˆ’ 1 ) = 0
  • C π‘₯ ( π‘₯ + 2 ) + π‘˜ = 0
  • D π‘₯ ( π‘₯ + 2 ) = 0
  • E π‘₯ ( π‘₯ βˆ’ 1 ) = π‘˜

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 ( βˆ’ 1 , 2 )
  • B ( 0 , 0 )
  • C ( 3 , 3 0 )
  • D ( 1 , 0 )
  • E ( 2 , 8 )

Q2:

Use technology to plot the graph of 𝑓 ( π‘₯ ) = π‘₯ + 5 π‘₯ βˆ’ 1 0 0 3 2 , and use this graph to find the solutions to the equation π‘₯ + 5 π‘₯ = 1 0 0 3 2 to two decimal places.

  • A π‘₯ = 3 . 4 4
  • B π‘₯ = βˆ’ 3 . 4 4
  • C π‘₯ = βˆ’ 1 2 . 8 1
  • D π‘₯ = βˆ’ 7 . 0 3
  • E π‘₯ = 7 . 0 3

Q3:

Use technology to plot the graph of 𝑓 ( π‘₯ ) = π‘₯ + 3 π‘₯ βˆ’ 2 3 2 , and use this graph to find the solutions to the equation π‘₯ + 3 π‘₯ = 2 3 2 to two decimal places.

  • A π‘₯ = βˆ’ 2 . 7 3 , π‘₯ = βˆ’ 1 , π‘₯ = 0 . 7 3
  • B π‘₯ = βˆ’ 0 . 5 6 , π‘₯ = 0 , π‘₯ = 3 . 5 6
  • C π‘₯ = βˆ’ 0 . 7 3 , π‘₯ = 1 , π‘₯ = 2 . 7 3
  • D π‘₯ = βˆ’ 3 . 5 6 , π‘₯ = 0 , π‘₯ = 0 . 5 6
  • E π‘₯ = βˆ’ 2 . 0 0 , π‘₯ = 1 . 0 0 , π‘₯ = 0 . 0 0
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