Complex Numbers

Lesson

To be able to write some numbers in complex form a little algebraic manipulation may be necessary, mostly involving the fact that $\sqrt{-1}=i$√−1=`i` or that $-1=i^2$−1=`i`2.

When we combine this algebraic manipulation with the use of our quadratic manipulations using the Quadratic Formula or Completing the Square, we can find both real and complex solutions to quadratic equations.

The same premise applies as when you are using the quadratic formula or completing the square factorisation processes. However when simplifying the final answer, if the discriminant is negative, then we employ the complex number simplification processes.

Let's look at some examples:

Solve $x^2-2x+3=0$`x`2−2`x`+3=0, using the quadratic formula

$x$x |
$=$= | $\frac{-b\pm\sqrt{b^2-4ac}}{2a}$−b±√b2−4ac2a |

$x$x |
$=$= | $\frac{-\left(-2\right)\pm\sqrt{\left(-2\right)^2-4\left(1\right)\left(3\right)}}{2\left(1\right)}$−(−2)±√(−2)2−4(1)(3)2(1) |

$x$x |
$=$= | $\frac{2\pm\sqrt{4-12}}{2}$2±√4−122 |

$x$x |
$=$= | $\frac{2\pm\sqrt{-8}}{2}$2±√−82 |

$x$x |
$=$= | $\frac{2\pm2\sqrt{-2}}{2}$2±2√−22 |

$x$x |
$=$= | $1\pm\sqrt{-2}$1±√−2 |

$x$x |
$=$= | $1\pm\sqrt{-1}\sqrt{2}$1±√−1√2 |

$x$x |
$=$= | $1\pm i\sqrt{2}$1±i√2 |

$x$x |
$=$= | $1\pm\sqrt{2}i$1±√2i |

Solve the same quadratic, $x^2-2x+3=0$`x`2−2`x`+3=0, using the method of completing the square.

$x^2-2x+3$x2−2x+3 |
$=$= | $0$0 |

$x^2-2x+(-1)^2+3-(-1)^2$x2−2x+(−1)2+3−(−1)2 |
$=$= | $0$0 |

$\left(x-1\right)^2+2$(x−1)2+2 |
$=$= | $0$0 |

$\left(x-1\right)^2$(x−1)2 |
$=$= | $-2$−2 |

$x-1$x−1 |
$=$= | $\pm\sqrt{-2}$±√−2 |

$x$x |
$=$= | $1\pm\sqrt{-2}$1±√−2 |

$x$x |
$=$= | $1\pm\sqrt{2}i$1±√2i |

Activity 1

_{Try this yourself before going to look at my solution.}

This is something you may or may not have done before, but as an algebraic exercise its very worthwhile.

Show the algebraic process for completing the square of

$ax^2+bx+c=0$`a``x`2+`b``x`+`c`=0

Then state the roots and possible cases.

(see here for the solution)

Activity 2

^{Try this for yourself before you check out the solution.}

Given that $p$`p` and $q$`q` are real and that $1+2i$1+2`i` is a root of the equation

$z^2+\left(p+5i\right)z+q\left(2-i\right)=0$`z`2+(`p`+5`i`)`z`+`q`(2−`i`)=0

Determine:

a) The values of $p$`p` and $q$`q`

b) The other root of the equation

(see here for the solution)

Below is the result after using the quadratic formula to solve an equation.

$x=\frac{-3\pm\sqrt{9-4\times56}}{16}$`x`=−3±√9−4×5616

Which of the following is true?

The equation has one real solution.

AThe equation has two real solutions.

BThe equation has complex solutions.

CThe equation has one real solution.

AThe equation has two real solutions.

BThe equation has complex solutions.

C

Which of the following equations would have non-real solutions?

$-4\left(x-5\right)^2=8$−4(

`x`−5)2=8A$4\left(x+5\right)^2=8$4(

`x`+5)2=8B$4\left(x+5\right)^2=-8$4(

`x`+5)2=−8C$-4\left(x-5\right)^2=-8$−4(

`x`−5)2=−8D$-4\left(x-5\right)^2=8$−4(

`x`−5)2=8A$4\left(x+5\right)^2=8$4(

`x`+5)2=8B$4\left(x+5\right)^2=-8$4(

`x`+5)2=−8C$-4\left(x-5\right)^2=-8$−4(

`x`−5)2=−8D

Solve $4x^2+5x+2=0$4`x`2+5`x`+2=0, stating your solutions in the form $a\pm bi$`a`±`b``i`.

Manipulate complex numbers and present them graphically

Apply the algebra of complex numbers in solving problems