The Quadratic Formula and Complex Solutions: IM Alg2.3.18
[url=https://im.kendallhunt.com/HS/teachers/3/3/18/preparation.html]“The Quadratic Formula and Complex Solutions”[/url] from IM Algebra II © 2019 [url=https://www.illustrativemathematics.org/]Illustrative Mathematics[/url]. Licensed under the [url=https://creativecommons.org/licenses/by/4.0/]Creative Commons Attribution 4.0[/url] license.
Table of Contents
- Alg2.3.18 Lesson
- IM Alg2.3.18 Lesson: The Quadratic Formula and Complex Solutions
- Alg2.3.18 Practice
- IM Alg2.3.18 Practice: The Quadratic Formula and Complex Solutions
IM Alg2.3.18 Lesson: The Quadratic Formula and Complex Solutions
Mentally decide whether the solutions to each equation are real numbers or not.
[math]w^2=\text{-}367[/math]
[math]x^2+25=0[/math]
[math](y+5)^2=0[/math]
[math](z+5)^2=\text{-}367[/math]
[size=150]Kiran was using the quadratic formula to solve the equation [math]x^2-12x+41=0[/math]. [br]He wrote this:[br][center][math]x=\frac{12\pm\sqrt{144-164}}{2}[/math][/center]Then he noticed that the number inside the square root is negative and said, “This equation doesn’t have any solutions.”[br][/size][br]Do you agree with Kiran? Explain your reasoning.
Write [math]\sqrt{\text{-}20}[/math] as an imaginary number.
Solve the equation 3x²-10x+50=0 and plot the solutions in the complex plane.
[size=150]Although imaginary numbers let us describe complex solutions to quadratic equations, they were actually discovered and accepted because they could help us find real solutions to equations with polynomials of degree 3. In the 16th century, mathematicians discovered a cubic formula for solving equations of degree 3, but to use it they sometimes had to work with complex numbers. Let’s see an example where this comes up.[/size][br][br]To find a solution to the equation [math]x^3-px-q=0[/math] the cubic formula would first tell us to find a complex number, [math]z[/math], which is [math]\frac{q}{2}+i\sqrt{(\frac{p}{3})^3-(\frac{q}{2})^2}[/math]. Find [math]z[/math] when our equation is [math]x^3-15x-4=0[/math].[br]
The next step is to find a complex number [math]w[/math] so that [math]w^3=z[/math]. Show that [math]w=2+i[/math] works for the [math]z[/math] we found in step 1.[br]
If we write [math]w=a+b[/math]i where [math]a[/math] and [math]b[/math] are real numbers, the solutions to our equation are [math]2a[/math], [math]-a+b\sqrt{3}[/math], and [math]-a-b\sqrt{3}[/math]. What are the three solutions to our equation [math]x^3-15x-a=0[/math]?[br]
For each row, you and your partner will each solve a quadratic equation. You should each get the same answer. If you disagree, work to reach agreement.
IM Alg2.3.18 Practice: The Quadratic Formula and Complex Solutions
[size=150]Clare solves the quadratic equation [math]4x^2+12x+58=0[/math], but when she checks her answer, she realizes she made a mistake. Explain what Clare's mistake was.[/size][br][br][math]x=\frac{\text{-}12\pm\sqrt{12^2-4\cdot4\cdot58}}{2\cdot4}[/math][br][math]x=\frac{\text{-}12\pm\sqrt{144-928}}{8}[/math][br][math]x=\frac{\text{-}12\pm\sqrt{\text{-}784}}{8}[/math][br][math]x=\frac{\text{-}12\pm28i}{8}[/math][br][math]x=\text{-}1.5\pm28i[/math]
[size=150]Write in the form [math]a+bi[/math], where [math]a[/math] and [math]b[/math] are real numbers:[/size][br][br][math]\frac{5\pm\sqrt{\text{-}4}}{3}[/math]
[math]\frac{10\pm\sqrt{\text{-}16}}{2}[/math]
[math]\frac{\text{-}3\pm\sqrt{\text{-}144}}{6}[/math]
Priya is using the quadratic formula to solve two different quadratic equations.
[size=150]For the first equation, she writes [math]x=\frac{4\pm\sqrt{16-72}}{12}[/math][br]For the second equation, she writes [math]x=\frac{8\pm\sqrt{64-24}}{6}[/math][/size][br][br]Which equation(s) will have real solutions? Which equation(s) will have non-real solutions? Explain how you know.
Find the exact solution(s) to each of these equations, or explain why there is no solution.
[math]x^2=25[/math]
[math]x^3=27[/math]
[math]x^2=12[/math]
[math]x^3=12[/math]
[size=150]Kiran is solving the equation [math]\sqrt{x+2}-5=11[/math] and decides to start by squaring both sides. Which equation results if Kiran squares both sides as his first step?[/size]