Completing the Square and Complex Solutions: IM Alg2.3.17
[url=https://im.kendallhunt.com/HS/teachers/3/3/17/preparation.html]“Completing the Square 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
- Alg2.3.17 Lesson
- IM Alg2.3.17 Lesson: Completing the Square and Complex Solutions
- Alg2.3.17 Practice
- IM Alg2.3.17 Practice: Completing the Square and Complex Solutions
IM Alg2.3.17 Lesson: Completing the Square and Complex Solutions
Match each equation in standard form to its factored form and its solutions.
What are the solutions to these equations?
[math](x-5)^2=0[/math]
[math](x-5)^2=1[/math]
[math](x-5)^2=\text{-}1[/math]
Solve these equations by completing the square.
[math]x^2-8x+13=0[/math]
[math]x^2-8x+19=0[/math]
[size=150]For which values of[math]a[/math] does the equation [math]x^2-8x+a=0[/math] have two real solutions? One real solution? No real solutions? Explain your reasoning.[/size]
How many real solutions does each equation have? How many non-real solutions?
[math]x^2-8x+13=0[/math]
[math]x^2-8x+16=0[/math]
[math]x^2-8x+19=0[/math]
How do the graphs of these functions help us answer the previous question?
[math]f(x)=x^2-8x+13[/math][br][math]g(x)=x^2-8x+16[/math][br][math]h(x)=x^2-8x+19[/math]
Use the applet if needed
IM Alg2.3.17 Practice: Completing the Square and Complex Solutions
Find the solution or solutions to each equation.
[math]x^2+0.5x-14=0[/math]
[math]x^2+12x+36=0[/math]
[math]x^2-3x+8=0[/math]
[math]x^2+4=0[/math]
[size=150]Which describes the solutions to the equation [math]x^2+7=0[/math]?[/size]
[size=150]Explain how you know [math]\sqrt{3x+2}=\text{-}16[/math] has no solutions.[/size]
Here is an applet of some graphs.
Using the applet above, determine the number of real solutions and non-real solutions to each equation. Use the graphs; don't do any calculations to find the solutions.
[math]x^2-6x+7=0[/math]
[math]3x^2+2x+1=0[/math]
[math]\text{-}x^2-3x+2=0[/math]
[math]x^2-6x+7=\text{-}2[/math]
[math]\text{-}x^2-3x+2=6[/math]
[math]3x^2+2x+1=2[/math]
[size=150]Write [math](5-5i)^2[/math] in the form [math]a+bi[/math], where [math]a[/math] and [math]b[/math] are real numbers.[br][/size]
[size=150]Write [math](5-5i)^4[/math] in the form [math]a+bi[/math], where [math]a[/math] and [math]b[/math] are real numbers.[br][/size]
What number n makes this equation true?
[math]x^2+11x+\frac{121}{4}=(x+n)^2[/math]