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

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

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

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

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

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

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

#### Example 2

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

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

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

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

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

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

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

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

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

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

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

#### Example 3

(Graphing calculator exercise)

A graphing calculator can perform operations with complex numbers. Press mode. Scroll down and select a+bi\begin{align*}a + bi\end{align*}. 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. (45i)(3+2i)\begin{align*}(4 - 5i) - (3 + 2i)\end{align*}

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

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

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

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

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

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

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

168i+79i\begin{align*}16 - 8i + 7 - 9i\end{align*} Distribute the negative

2317i\begin{align*}23 - 17i\end{align*} Collect like terms and simplify

#### Example 4

Subtract the complex numbers.

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

(9+7i)+(63i)\begin{align*}(9 + 7i) + (-6 -3i)\end{align*} Distribute the negative

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

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

1. (18+81)(1864)\begin{align*}(18 + \sqrt{-81}) - (18 - \sqrt{-64})\end{align*}

(18+9i)(188i)\begin{align*}(18 + 9i) - (18 - 8i)\end{align*} Simplify the roots

18+9i18+8i\begin{align*}18 + 9i - 18 + 8i\end{align*} Distribute the negative

1818+9i+8i\begin{align*}18 - 18 + 9i + 8i\end{align*} Group real and imaginary parts

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

#### Example 5

Solve the equations and express them as complex numbers.

1. 18x22x+24=0\begin{align*}18x^2 - 2x + 24 = 0\end{align*}

9x2x+12=0\begin{align*}9x^2 - x + 12 = 0\end{align*} Divide both sides by 2

A=9|B=1|C=12\begin{align*}A = 9 | B = -1 | C = 12\end{align*} Identify A, B, and C using standard form: Ax2+Bx+C=0\begin{align*}Ax^2 + Bx + C = 0\end{align*}

1±14(9)(12)2(9)\begin{align*}\frac{1 \pm \sqrt{1 - 4 (9)(12)}}{2(9)}\end{align*} Substitute the terms into the quadratic formula B±B24AC2A\begin{align*}\frac{-B \pm \sqrt{B^2 - 4 A C}}{2A}\end{align*}

x=1±i43118\begin{align*}x = \frac{1 \pm i\sqrt{431}}{18}\end{align*} By the quadratic formula

1. 1245x2=1425x1115\begin{align*}12\frac{4}{5}x^2 = 14\frac{2}{5}x - 11\frac{1}{5}\end{align*}

645x2725x+565=0\begin{align*}\frac{64}{5}x^2 - \frac{72}{5}x + \frac{56}{5} = 0\end{align*} Convert to improper fractions

64x272x+56=0\begin{align*}64x^2 - 72x + 56 = 0\end{align*} Multiply both sides by 5

8x29x+7=0\begin{align*}8x^2 -9x +7 = 0\end{align*} Divide both sides by 8

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

x=9±i14316\begin{align*}x = \frac{9 \pm i\sqrt{143}}{16}\end{align*} By the quadratic formula

### Review

Add the complex numbers.

1. (4+4i)+(3+6i)\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*}

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

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### Vocabulary Language: English

TermDefinition
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.