# Solving Cubic Equations

## Solving Cubic Equations Revision

**Solving Cubic Equations**

Solving higher order equations such as cubics can pose many challenges, the normal way of dealing with these is via **trial and improvement**.

**Trial and Improvement Process**

**Trial and improvement** is the process of taking an educated guess, seeing what result you get, and then improving on your previous guess with a new one.

**Example: **The solution to the equation x^3-6x=72 lies between 4 and 5. Use trial and improvement to find a solution correct to one decimal place.

**Step 1:** Make an educated guess, here choose x=4.5 since this is the middle of our range. This the first “trial”.

**Step 2: **Substitute the trial into the equation

(4.5)^3-6(4.5)=64.125

**Step 3: **Improve on the previous trial, here choose x=4.6 since the last result was too small.

(4.6)^3-6(4.6)=69.736

This result was too small.

**Step 4: **Repeat step 3 until you get the opposite outcome, in this case look for a result that is too big. The table shows the various trials and conclusions from each trial.

**Step 5:** Decide which solution to use. Since 4.6 was too small and 4.7 was too big, we must test 4.65 as this is halfway in between.

(4.65)^3-6(4.65)=72.644625

Since this is too big, we must choose x=4.6 as the solution.

## Solving Cubic Equations Example Questions

**Question 1: **The solution to the equation 2x^3+7x=7 lies between 0 and 1. Use trial and improvement to find a solution correct to one decimal place.

**[3 marks]**

Start at 0.5 since this is the middle of the range.

Create a table of results

x | Outcome | Conclusion |

0.5 | 3.75 | Too small |

0.6 | 4.632 | Too small |

0.7 | 5.586 | Too small |

0.8 | 6.624 | Too small |

0.9 | 7.758 | Too big |

Since 0.8 is too small and 0.9 is too big, we must trial 0.85

2(0.85)^3+7(0.85)=7.17825Since this is too big, we choose x=0.8 as the solution.

**Question 2: **The solution to the equation x^3-11x+15=0 lies between -3 and -4. Use trial and improvement to find a solution correct to one decimal place.

**[3 marks]**

Start at -3.5 since this is the middle of the range.

Create a table of results

x | Outcome | Conclusion |

-3.5 | 10.625 | Too big |

-3.6 | 7.944 | Too big |

-3.7 | 5.047 | Too big |

-3.8 | 1.928 | Too big |

-3.9 | -1.419 | Too small |

Since -3.9 is too small and -3.8 is too big, we must trial -3.85

(-3.85)^3+11(-3.85)+15=0.283375Since this is too big, we choose x=-3.9 as the solution.

**Question 3: **The equation 3x^3+x^2-13x+6=0 has a solution between 1 and 2. Use trial and improvement to find a solution correct to one decimal place.

**[3 marks]**

Start at 1.5 since this is the middle of the range.

Create a table of results

x | Outcome | Conclusion |

1.5 | -1.125 | Too small |

1.6 | 0.048 | Too big |

Since 1.5 is too small and 1.6 is too big, we must trial 1.55

3(1.55)^3+(1.55)^2-12(1.55)+6=-0.575875Since this is too small, we choose x=1.6 as the solution.