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Quadratic Formula and Complex Sums

Conversion between a + bi and (a, b), (r, theta), and rcistheta.

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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.

Quadratic Formula and Complex Sums

The Quadratic Formula and the Discriminant

If ax2 + bx + c = 0

then \begin{align*}x = \frac{-b \pm \sqrt{b^2 - 4ac}} {2a}\end{align*}x=b±b24ac2a

Recall that b2 - 4ac is called the discriminant.

If b2 - 4ac > 0 then there are two unequal real solutions.

If b2 - 4ac = 0 then there are two equal real solutions.

If b2 - 4ac < 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

  

 

Examples

Example 1

Combine the complex numbers using addition or subtraction.

  1. \begin{align*}(5 + 3i) + (6 - 8i)\end{align*}(5+3i)+(68i)

Applying the commutative property: \begin{align*}(5 + 6) + (3i + -8i) = 11 - 5i\end{align*}(5+6)+(3i+8i)=115i

  1. \begin{align*}(3 - 2i) - (2 - 4i)\end{align*}(32i)(24i)

Distribute the negative: \begin{align*}(3 - 2i) + (-2 +4i)\end{align*}(32i)+(2+4i), then apply the commutative property: \begin{align*}(3 - 2) + (-2i + 4i) = 1 + 2i\end{align*}(32)+(2i+4i)=1+2i

  1. \begin{align*}(6) + (4 - 3i)\end{align*}(6)+(43i)

The imaginary coefficient in the first term is 0, so applying commutative property gives: \begin{align*}(6 + 4) + (0i - 3i) = 10 - 3i\end{align*}(6+4)+(0i3i)=103i

Example 2

Given: \begin{align*}x^2 + 4x + 6 = 0\end{align*}x2+4x+6=0.

\begin{align*}x^2 + 4x + 6 = 0\end{align*}x2+4x+6=0

\begin{align*}a = 1, b = 4, c = 6\end{align*}a=1,b=4,c=6

\begin{align*}x = \frac{-4 \pm \sqrt{(4)^2 - (4)(1)(6)}} {2(1)}\end{align*}x=4±(4)2(4)(1)(6)2(1)

  1. Use the discriminant to predict the nature of the roots.

Since \begin{align*}b^2 - 4ac = -8\end{align*}b24ac=8, there will be 2 complex solutions (no real solutions)

\begin{align*}x = \frac{-4 \pm \sqrt{-8}} {2}\end{align*}x=4±82

  1. Use the quadratic formula to solve and identify the roots.

\begin{align*}x = \frac{-4 \pm 2i \sqrt{2}} {2}\end{align*}x=4±2i22

  1. Express the roots as complex numbers in standard form.

\begin{align*}x = -2 \pm i \sqrt{2}\end{align*}x=2±i2

Example 3

(Graphing calculator exercise)

A graphing calculator can perform operations with complex numbers. Press mode. Scroll down and select \begin{align*}a + bi\end{align*}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.

Add or subtract the complex numbers using a graphing calculator:

  1. \begin{align*}(4 - 5i) - (3 + 2i)\end{align*}(45i)(3+2i)

\begin{align*}1 - 7i\end{align*}17i

  1. \begin{align*}(3 - 7i) + (2 + i)\end{align*}(37i)+(2+i)

\begin{align*}5 - 6i\end{align*}56i

  1. \begin{align*}(- 2i) + (2 + 6i)\end{align*}(2i)+(2+6i)

\begin{align*}2 + 4i\end{align*}2+4i

  1.  \begin{align*}(16 - \sqrt{-64}) + (7 - \sqrt{-81})\end{align*}(1664)+(781)

\begin{align*}(16 - 8i) + (7 - 9i)\end{align*}(168i)+(79i) Simplify the roots in terms of i

\begin{align*}16 - 8i + 7 - 9i\end{align*} Distribute the negative

\begin{align*}23 - 17i\end{align*} Collect like terms and simplify

Example 4

Subtract the complex numbers.

  1. \begin{align*}(9 + 7i) - (6 + 3i)\end{align*}

\begin{align*}(9 + 7i) + (-6 -3i)\end{align*} Distribute the negative

\begin{align*}(9 + -6) + (7i + - 3i)\end{align*} Group the real part and the imaginary part of each

\begin{align*}3 + 4i\end{align*} Combine like terms

  1. \begin{align*}(18 + \sqrt{-81}) - (18 - \sqrt{-64})\end{align*}

\begin{align*}(18 + 9i) - (18 - 8i)\end{align*} Simplify the roots

\begin{align*}18 + 9i - 18 + 8i\end{align*} Distribute the negative

\begin{align*}18 - 18 + 9i + 8i\end{align*} Group real and imaginary parts

\begin{align*}17i\end{align*} Simplify

Example 5

Solve the equations and express them as complex numbers.

  1. \begin{align*}18x^2 - 2x + 24 = 0\end{align*}

\begin{align*}9x^2 - x + 12 = 0\end{align*} Divide both sides by 2

\begin{align*}A = 9 | B = -1 | C = 12\end{align*} Identify A, B, and C using standard form: \begin{align*}Ax^2 + Bx + C = 0\end{align*}

\begin{align*}\frac{1 \pm \sqrt{1 - 4 (9)(12)}}{2(9)}\end{align*} Substitute the terms into the quadratic formula \begin{align*}\frac{-B \pm \sqrt{B^2 - 4 A C}}{2A}\end{align*}

\begin{align*}x = \frac{1 \pm i\sqrt{431}}{18}\end{align*} By the quadratic formula

  1. \begin{align*}12\frac{4}{5}x^2 = 14\frac{2}{5}x - 11\frac{1}{5}\end{align*}

\begin{align*}\frac{64}{5}x^2 - \frac{72}{5}x + \frac{56}{5} = 0\end{align*} Convert to improper fractions

\begin{align*}64x^2 - 72x + 56 = 0\end{align*} Multiply both sides by 5

\begin{align*}8x^2 -9x +7 = 0\end{align*} Divide both sides by 8

\begin{align*}A = 8 | B = -9 | C = 7\end{align*} Extract values for the quadratic formula

\begin{align*}x = \frac{9 \pm i\sqrt{143}}{16}\end{align*} By the quadratic formula

      

 

Review

Add the complex numbers.

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

Subtract the complex numbers.

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

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

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

Review (Answers)

To see the Review answers, open this PDF file and look for section 4.6. 

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Vocabulary

complex number

A complex number is the sum of a real number and an imaginary number, written in the form a + bi.

complex root

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

discriminant

The discriminant is the part of the quadratic formula under the radical, b^2 - 4ac. A positive discriminant suggests two real roots to the quadratic equation, a zero suggests one real root with multiplicity two, and a negative indicates two complex roots.

Imaginary Numbers

An imaginary number is a number that can be written as the product of a real number and i.

Quadratic Formula

The quadratic formula states that for any quadratic equation in the form ax^2+bx+c=0, x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}.

Real Number

A real number is a number that can be plotted on a number line. Real numbers include all rational and irrational numbers.

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