Solving Quadratic Equations (Factored Form): IM Alg1.7.9
[url=https://im.kendallhunt.com/HS/teachers/1/7/9/preparation.html]“Solving Quadratic Equations by Using Factored Form”[/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
- Alg1.7.9 Lesson
- IM Alg1.7.9 Lesson: Solving Quadratic Equations by Using Factored Form
- Alg1.7.9 Practice
- IM Alg1.7.9 Practice: Solving Quadratic Equations by Using Factored Form
IM Alg1.7.9 Lesson: Solving Quadratic Equations by Using Factored Form
[size=150]Let's try to find at least one solution to [math]x^2-2x-35=0[/math][/size].[br][br]Choose a whole number between 0 and 10.
Evaluate the expression [math]x^2-2x-35[/math], using your number for [math]x[/math].
[size=100]If your number doesn't give a value of 0, look for someone in your class who may have chosen a number that does make the expression equal 0.[/size] Which number is it?
[size=100]There is another number that would make the expression [math]x^2-2x-35[/math] equal 0. [/size][br]Can you find it?
[size=150]To solve the equation [math]n^2-2n=99[/math], Tyler wrote out the following steps. Analyze Tyler’s work. Write down what Tyler did in each step.[/size] [br][br][math]\begin {align} n^2-2n&= 99 &\qquad&\text{Original equation}\\\\n^2-2n-99&=0 &\qquad &\text{Step 1}\\\\ (n-11)(n+9)&=0 &\qquad&\text{Step 2} \\\\ n-11=0 \quad \text{or} \quad &n+9=0 &\qquad& \text{Step 3}\\\\ n=11 \quad \text{or} \quad &n=\text-9 &\qquad&\text{Step 4} \end {align}[/math]
[size=150]Solve each equation by rewriting it in factored form and using the zero product property. Show your reasoning.[br][br][math]x^2+8x+15=0[/math] [br][/size]
[math]x^2-8x+12=5[/math]
[math]x^2-10x-11=0[/math]
[math]49-x^2=0[/math]
[math](x+4)(x+5)-30=0[/math]
Solve this equation and explain or show your reasoning.[br][br][math](x^2-x-20)(x^2+2x-3)=(x^2+2x-8)(x^2-8x+15)[/math]
The other day, we saw that a quadratic equation can have 0, 1, or 2 solutions. Sketch the graph that represents the quadratic function: one that has no zeros.
Sketch the graph that represent the quadratic function: one with 1 zeros.
Sketch the graph that represent the quadratic function: one with 2 zeros.
Use graphing technology to graph the function defined by f(x)=x²-2x+1.
What do you notice about the [math]x[/math]-intercepts of the graph?
What do the [math]x[/math]-intercepts reveal about the function?
[size=100]Solve [math]x^2-2x+1=0[/math] by using the factored form and zero product property.[/size][br]Show your reasoning. What solutions do you get?
Write an equation to represent another quadratic function that you think will only have one zero.
Graph it to check your prediction.
IM Alg1.7.9 Practice: Solving Quadratic Equations by Using Factored Form
[size=150]Find [b]all[/b] the solutions to each equation.[/size][br][br][math]x(x-1)=0[/math]
[math](5-x)(5+x)=0[/math]
[math](2x+1)(x+8)=0[/math]
[math](3x-3)(3x-3)=0[/math]
[math](7-x)(x+4)=0[/math]
Rewrite each equation in factored form and solve using the zero product property.
[math]d^2-7d+6=0[/math]
[math]x^2+18x+81=0[/math]
[math]u^2+7u-60=0[/math]
[math]x^2+0.2x+0.01=0[/math]
[size=150]Here is how Elena solves the quadratic equation [math]x^2-3x-18=0[/math].[/size][br][br][math]\displaystyle \begin{align} x^2 -3x -18 &=0\\ (x-3)(x+6)&=0\\ x-3=0 \quad \text { or } &\quad x+6=0\\ x=3\quad \text{ or } &\quad x= \text- 6\\ \end{align}\\[/math][br][br]Is her work correct? If you think there is an error, explain the error and correct it.[br][br]Otherwise, check her solutions by substituting them into the original equation and showing that the equation remains true.
Jada is working on solving a quadratic equation, as shown here.
[table][tr][td][math]\begin{align} p^2-5p&=0\\p(p-5)&=0\\p-5&=0\\p&=5\end{align}[/math][br][/td][td][size=150]She thinks that her solution is correct because substituting 5 for [math]p[/math] in [br]the original expression [math]p^2-5p[/math] gives [math]5^2-5(5)[/math], which is [math]25-25[/math] or 0.[/size][/td][/tr][/table][br]Explain the mistake that Jada made and show the correct solutions.
[size=150]Choose a statement to correctly describe the zero product property. [br][br]If [math]a[/math] and [math]b[/math] are numbers, and [math]a\cdot b=0[/math], then:[/size]
[size=150] Which expression is equivalent to [math]x^2-7x+12[/math]?[/size][br]
These quadratic expressions are given in standard form. Rewrite each expression in factored form. If you get stuck, try drawing a diagram in the applet below.
[math]x^2+7x+6[/math]
[math]x^2-7x+6[/math]
[math]x^2-5x+6[/math]
[math]x^2+5x+6[/math]
[size=150]Select [b]all[/b] the functions whose output values will eventually overtake the output values of function [math]f[/math] defined by [math]f(x)=25x^2[/math].[/size]
A piecewise function, p, is defined by this rule:
[math]p(x)=\begin{cases} x-1, \quad x\leq \text- 2 \\ 2x-1,\quad x>\text{-}2\\ \end{cases}[/math][br][br][size=150]Find the value of [math]p[/math] at each given input.[br][br][math]p(\text{-}20)[/math] [br][/size]
[math]p(\text{-}2)[/math]
[math]p(4)[/math]
[math]p(5.7)[/math]