Completing the Square (Part 2): IM Alg1.7.13
[url=https://im.kendallhunt.com/HS/teachers/1/7/13/preparation.html]“Completing the Square (Part 2)”[/url] from IM Algebra I © 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
- Alg1.7.13 Lesson
- IM Alg1.7.13 Lesson: Completing the Square (Part 2)
- Alg1.7.13 Practice
- IM Alg1.7.13 Practice: Completing the Square (Part 2)
IM Alg1.7.13 Lesson: Completing the Square (Part 2)
Solve each equation mentally.
[math]x+x=\frac{1}{4}[/math]
[math]\left(\frac{3}{2}\right)^2=x[/math]
[math]\frac{3}{5}+x=\frac{9}{5}[/math]
[math]\frac{1}{12}+x=\frac{1}{4}[/math]
Solve these equations by completing the square.
[math]\left(x-3\right)\left(x+1\right)=5[/math]
[math]x^2+\frac{1}{2}x=\frac{3}{16}[/math]
[math]x^2+3x+\frac{8}{4}=0[/math]
[math]\left(7-x\right)\left(3-x\right)+3=0[/math]
[math]x^2+1.6x+0.63=0[/math]
Show that the equation [math]x^2+10x+9=0[/math] is equivalent to [math]\left(x+3\right)^2+4x=0[/math].[br]
Write an equation that is equivalent to [math]x^2+9x+16=0[/math] and that includes [math]\left(x+4\right)^2[/math].[br]
Does this method help you find solutions to the equations? Explain your reasoning.[br]
Here are four equations, followed by worked solutions of the equations. Each solution has at least one error.
[list][size=150][*]Solve one or more of these equations by completing the square.[/*][*]Then, look at the worked solution of the same equation as the one you solved. Find and describe the error or errors in the worked solution.[/*][/size][/list][math]x^2+14x=-24[/math]
Worked solution (with errors):[br][math]\displaystyle \begin {align} x^2 + 14x &= \text-24\\ x^2 + 14x + 28 &= 4\\ (x+7)^2 &= 4\\ \\x+7 = 2 \quad &\text {or} \quad x+7 = \text-2\\ x = \text-5 \quad &\text {or} \quad x = \text-9 \end{align}[/math]
[math]x^2-10x+16=0[/math]
Worked solution (with errors):[br][math]\displaystyle \begin {align} x^2 - 10x + 16 &= 0\\x^2 - 10x + 25 &= 9\\(x - 5)^2 &= 9\\ \\x-5=9 \quad &\text {or} \quad x-5 = \text-9\\ x=14 \quad &\text {or} \quad x=\text-4 \end{align}[/math]
[math]x^2+2.4x=-0.8[/math]
Worked solution (with errors):[br][size=150][size=100][math]\displaystyle \begin {align}x^2 + 2.4x &= \text-0.8\\x^2 + 2.4x + 1.44 &= 0.64\\(x + 1.2)^2&=0.64\\x+1.2 &= 0.8\\ x &=\text -0.4 \end{align}[/math][/size][/size]
[math]x^2-\frac{6}{5}x+\frac{1}{5}=0[/math]
Worked solution (with errors):[br][math]\displaystyle \begin {align} x^2 - \frac65 x + \frac15 &= 0\\x^2 - \frac65 x + \frac{9}{25} &= \frac{9}{25}\\ \left(x-\frac35\right)^2 &= \frac{9}{25}\\ \\x-\frac35= \frac35 \quad &\text {or} \quad x-\frac35=\text- \frac35\\ x=\frac65 \quad &\text {or} \quad x=0 \end{align}[/math]
IM Alg1.7.13 Practice: Completing the Square (Part 2)
Add the number that would make the expression a perfect square. Next, write an equivalent expression in factored form.
[math]x^2+3x[/math]
[math]x^2+0.6x[/math]
[math]x^2-11x[/math]
[math]x^2-\frac{5}{2}x[/math]
[math]x^2+x[/math]
[size=150]Noah is solving the equation [math]x^2+8x+15=3[/math]. He begins by rewriting the expression on the left in factored form and writes [math]\left(x+3\right)\left(x+5\right)=3[/math]. He does not know what to do next. [br][br]Noah knows that the solutions are [math]x=-2[/math] and [math]x=6[/math], but is not sure how to get to these values [/size][size=150]from his equation. [br][br]Solve the original equation by completing the square. [/size]
[size=150]An equation and its solutions are given. Explain or show how to solve the equation by completing the square.[/size][br][br][math]x^2+20x+50=14[/math]. The solutions are [math]x=-18[/math] and [math]x=-2[/math].
[math]x^2+1.6x=0.36[/math]. The solutions are [math]x=-1.8[/math] and [math]x=0.2[/math].
[math]x^2-5x=\frac{11}{4}[/math]. The solutions are [math]x=\frac{11}{2}[/math] and [math]x=-\frac{1}{2}[/math].
Solve each equation.
[math]x^2-0.5x=0.5[/math]
[math]x^2+0.8x=0.09[/math]
[math]x^2+\frac{13}{8}x=\frac{56}{36}[/math]
Match each quadratic expression given in factored form with an equivalent expression in standard form. One expression in standard form has no match.
[size=150]Four students solved the equation [math]x^2+225=0[/math]. Their work is shown here. Only one student solved it correctly.[/size][br][br][table][tr][td]Student A:[/td][td]Student B:[/td][/tr][tr][td][math]\displaystyle \begin{align} x^2 +225&=0\\ x^2&=\text -225\\ x=15 \quad &\text{ or } \quad x= \text- 15\\ \end{align}\\[/math][br][/td][td][math]\displaystyle \begin{align} x^2 +225&=0\\ x^2&=\text -225\\ \text{No} &\text{ solutions} \end{align}\\[/math][br][/td][/tr][tr][td]Student C:[/td][td]Student D:[/td][/tr][tr][td][math]\displaystyle \begin{align} x^2 +225&=0\\ (x-15)(x+15)&=0\\ x=15 \quad \text{ or } \quad x&= \text- 15\\ \end{align}\\[/math][/td][td][math]\displaystyle \begin{align} x^2 +225&=0\\ x^2&=225\\ x=15 \quad &\text{ or } \quad x= \text- 15\\ \end{align}\\[/math][/td][/tr][/table][br]Determine which student solved the equation correctly. For each of the incorrect solutions, explain the mistake.[br]