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Quadratic Formula and Complex Sums

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Quadratic Formula and Complex Sums

You probably remember when you first learned to use the Quadratic formula in Algebra and you ended up with a negative number under the root symbol. Chances are good that your instructor simply said something like "If you get a negative number under the root, there are no real answers, since there is no such thing as the root of a negative!"

Now that you are familiar with imaginary numbers, you can probably see that although it would have been an easy assumption at the time that "no real answers" just meant "no answers at all," that isn't true. "No real answers" may well mean that there ARE some "unreal" or imaginary answers.

Watch This

Embedded Video:

- James Sousa: Ex. 1: Adding and Subtracting Complex Numbers

Guidance

Review: The Quadratic Formula and the Discriminant

If ax 2 + bx + c = 0

then x = \frac{-b \pm \sqrt{b^2 - 4ac}} {2a}

Recall that b 2 - 4 ac is called the discriminant.

If b 2 - 4 ac > 0 then there are two unequal real solutions.

If b 2 - 4 ac = 0 then there are two equal real solutions.

If b 2 - 4 ac < 0 then there are two unequal complex solutions.

Sums and Differences of Complex Numbers

When adding (or subtracting) two or more complex numbers the fastest method is to add (or subtract) the real components to obtain the sum of the real numbers, and then separately add (or subtract) the imaginary coefficients to obtain the sum of the imaginary numbers, e.g.:

( a + bi ) + ( c + di ) = [ a + c ] + [ b + d ] i

Example A

Combine the complex numbers using addition or subtraction

a) (5 + 3i) + (6 - 8i)
b) (3 - 2i) - (2 - 4i)
c) (6) + (4 - 3i)

Solutions

a) Applying the commutative property: (5 + 6) + (3i + -8i) = 11 - 5i
b) Distribute the negative: (3 - 2i) + (-2 +4i) , then apply commutation: (3 - 2) + (-2i + 4i)  = 1 + 2i
c) The imaginary coefficient in the first term is 0 , so applying commutative property gives: (6 + 4) + (0i - 3i) = 10 - 3i

Example B

Given: x^2 + 4x + 6 = 0

a) Use the discriminant to predict the nature of the roots
b) Use the Quadratic Formula to solve and identify the roots
c) Express the roots as complex numbers in standard form

Solutions

x^2 + 4x + 6 = 0

a = 1, b = 4, c = 6

x = \frac{-4 \pm \sqrt{(4)^2 - (4)(1)(6)}} {2(1)}

a) Since b^2 - 4ac = -8 , there will be 2 complex solutions (no real solutions)

x = \frac{-4 \pm \sqrt{-8}} {2}

b) x = \frac{-4 \pm 2i \sqrt{2}} {2}
c) x = -2 \pm i \sqrt{2}

Example C

Graphing calculator exercise

Add or subtract the complex numbers using a graphing calculator:

a) (4 - 5i) - (3 + 2i)
b) (3 - 7i) + (2 + i)
c) (- 2i) + (2 + 6i)

Solution

A graphing calculator can perform operations with complex numbers. Press mode. Scroll down and select a + bi Press Quit. Now the calculator is able to perform operations with complex numbers in a + bi form. When the calculator is in complex number mode, be sure to use parenthesis to group the parts of the complex numbers.

Your answers should look like:

a) 1 - 7i
b) 5 - 6i
c) 2 + 4i

Vocabulary

The discriminant is the part of the Quadratic Formula under the radical: b^2 - 4ac . A positive discriminant suggests 2 real roots to the quadratic equation, a zero suggests 1 real root and 1 complex root, and a negative indicates 2 complex roots.

A complex root is a complex number that, when used as an input ( x ) value, results in an output ( y ) value of zero.

Guided Practice

1) Add the complex numbers

a) (3+ 4i) + (9 + 13i)
b) (16 - \sqrt{-64})  + (7 - \sqrt{-81})

2) Subtract the complex numbers

a) (9 + 7i) - (6 + 3i)
b) (18 + \sqrt{-81}) - (18 - \sqrt{-64})

3) Solve the equations and express them as complex numbers

a) 18x^2 - 2x + 24 = 0
b) 12\frac{4}{5}x^2 = 14\frac{2}{5}x - 11\frac{1}{5}

Solutions

1 a) To add the complex numbers (3+ 4i) + (9 + 13i)

(3 + 9)(4i + 13i) Group the real parts and the imaginary parts
12 + 17i Combine like terms and simplify
b) To add (16 - \sqrt{-64}) + (7 - \sqrt{-81})
(16 - 8i) + (7 - 9i) Simplify the roots in terms of i
16 - 8i + 7 - 9i Distribute the negative
23 - 17i Collect like terms and simplify

2 a) To subtract (9 + 7i) - (6 + 3i)

(9 + 7i) + (-6 -3i) Distribute the negative
(9 + -6) + (7i + - 3i) Group the real part and the imaginary part of each
3 + 4i Combine like terms
b) To subtract (18 + \sqrt{-81}) - (18 - \sqrt{-64})
(18 + 9i) - (18 - 8i) Simplify the roots
18 + 9i - 18 + 8i Distribute the negative
18 - 18 + 9i + 8i Group real and imaginary parts
17i Simplify

3 a) To solve 18x^2 - 2x + 24 = 0

9x^2 - x + 12 = 0 Divide both sides by 2
A = 9 | B = -1 | C = 12 Identify A, B, and C using standard form: Ax^2 + Bx + C = 0
\frac{1 \pm \sqrt{1 - 4 (9)(12)}}{2(9)} Substitute the terms into the Quadratic Formula \frac{-B \pm \sqrt{B^2 - 4 A C}}{2A}

x = \frac{1 \pm i\sqrt{431}}{18} By the Quadratic Formula

b) To solve 12\frac{4}{5}x^2 = 14\frac{2}{5}x - 11\frac{1}{5}

\frac{64}{5}x^2 - \frac{72}{5}x + \frac{56}{5} = 0 Convert to improper fractions
64x^2 - 72x + 56 = 0 Multiply both sides by 5
8x^2 -9x +7 = 0 Divide both sides by 8
A = 8 | B = -9 | C = 7 Extract values for the Quadratic Formula
x = \frac{9 \pm i\sqrt{143}}{16} By the Quadratic Formula

Practice

Add the complex numbers

  1. (4 + 4i) + (3 + 6i)
  2. (4 + 8i) + (8 + 8i)
  3. 20 \sqrt{-100} + \sqrt{-9}
  4. 3 \sqrt-{121} + 15 \sqrt{-81}
  5. -25\sqrt{-9} + 23\sqrt{-196}

Subtract Complex Numbers

  1. (9 + 8i) - (8 + 5i)
  2. (6 - 8i) - (12 + 9i)
  3. -5 \sqrt{-49} - 16 \sqrt{-1}
  4. 22 \sqrt{-144} - 9\sqrt{-4}
  5. 27 \sqrt {-196} - \sqrt{-1}

Solve each equation and express the result as a complex number.

  1. 12x\frac{4}{5}x = 3\frac{1}{5}x^2 + 16
  2. 3x^2 - 6x + 15 = 0
  3. 8x^2 - 5x + 11 = 0
  4. 34\frac{1}{2}x^2 - 23x + 19\frac{1}{6} = 0
  5. -36x^2 - 18x + 6 = 27
  6. When the sum of -4 + 8i and 2 - 9i is graphed, in which quadrant does it lie?
  7. If z_1 = -3 + 2i and z_2 = 4 - 3i , in which quadrant does the graph of (z_2 - z_1) lie?
  8. On a graph, if point A represents 2 - 3i and point B represents -2 - 5i , which quadrant contains A - B ?
  9. Find the sum of (-3 + 4i) and (-4 - 7i) and graph the result
  10. Graph the difference of ( 4 + 7i) and (2 - 3i)

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