Emma's copy of Using the Quadratic Formula to Solve Equations of the Form $ax^2 +bx+c=0$

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Emma Cliffe

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Emma Cliffe 7 years, 7 months ago

Created this as a copy of Using the Quadratic Formula to Solve Equations of the Form

$ax^2 +bx+c=0$ .

Name	Status	Author	Last Modified
Using the Quadratic Formula to Solve Equations of the Form $ax^2 +bx+c=0$	Ready to use	Hannah Aldous	07/01/2021 14:29
Use the quadratic formula to solve an equation in terms of an unknown variable	Ready to use	Hannah Aldous	20/11/2019 14:44
Katherine's copy of Using the Quadratic Formula to Solve Equations of the Form $ax^2 +bx+c=0$	draft	Katherine Tomlinson	02/08/2017 12:45
Emma's copy of Using the Quadratic Formula to Solve Equations of the Form $ax^2 +bx+c=0$	draft	Emma Cliffe	31/08/2017 12:59
Peter's copy of Using the Quadratic Formula to Solve Equations of the Form $ax^2 +bx+c=0$	draft	Peter Knowles	12/09/2017 12:39
Using the Quadratic Formula to Solve Equations of the Form $ax^2 +bx+c=0$ [L4 Randomised]	Needs to be tested	Matthew James Sykes	25/07/2018 11:06
Simon's copy of Using the Quadratic Formula to Solve Equations of the Form $ax^2 +bx+c=0$	draft	Simon Thomas	07/03/2019 09:55
Simon's copy of Use the quadratic formula to solve an equation in terms of an unknown variable	draft	Simon Thomas	07/03/2019 10:00
Use the quadratic formula to solve an equation in terms of an unknown variable	draft	Xiaodan Leng	11/07/2019 01:50
Solve Equations of the Form $ax^2 +bx+c=0$	draft	Thomas Waters	11/09/2019 22:18
Ugur's copy of Using the Quadratic Formula to Solve Equations of the Form $ax^2 +bx+c=0$	Ready to use	Ugur Efem	08/11/2021 16:44
Ugur's copy of Ugur's copy of Using the Quadratic Formula to Solve Equations of the Form $ax^2 +bx+c=0$	Ready to use	Ugur Efem	28/04/2022 11:08

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Question statement

When quadratic equations can't be factorised, or if equations are difficult to factorise (perhaps if the coefficients are large), we need to use the quadratic formula to solve the equations.

Use the quadratic formula to calculate values for $x$ in these equations. Input the possible values as $x_1$ and $x_2$ , where $x_1<x_2$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

			b	number	80
			c	number	64

			Name	Type	Generated Value

a1	integer	8
a2	integer	21
a3	integer	-28
a4	integer	9
b1	integer	6
b2	integer	27
b3	integer	-27
b4	integer	5
x1	number	-3.8319555463
p1	number	-4.6371006402
p2	number	0.9704339735
x2	number	1.2069555463
a	integer	15
m	integer	11

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
2c^0.5+a1^2

JME function reference

Description

Describe what this variable represents, and list any assumptions made about its value.

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Generated value: `number`

← Depends on:

This variable doesn't seem to be used anywhere.

Parts

Part a) Gap-fill
1. Gap Gap 0. Number entry
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2. Gap Gap 1. Number entry
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1. Step Information only
Part b) Gap-fill
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2. Gap Gap 1. Number entry
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Part c) Gap-fill
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Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

$\simplify{x^2+{a+m}x+{a*m}=0}$

$x_1=$ Gap 0.

$x_2=$ Gap 1.

Advice

The quadratic formula is

$x={\frac {-b\pm\sqrt{b^2-4\times a\times c}}{2a}}\text{.}$

a)

From the equation, we can read off values for $a$ , $b$ and $c$ :

$\begin{align} a&=1\text{,}\\ b&=\var{a+m}\text{,}\\ c&=\var{a*m} \text{.} \end{align}$

Substituting these values into the quadratic formula,

$x = \frac {-\var{a+m}\pm\sqrt{\var{a+m}^2-4\times \var{a*m}}}{2}\text{.}$

Note the $\pm$ symbol in the formula. This means there are two solutions: one using $+$ , the other using $-$ .

The two solutions are

$\begin{align} x_1&=\var{m}\text{,}\\ x_2&=\var{a}\text{.} \end{align}$

b)

Note that the right-hand side of the given equation is not zero. We need to rewrite it in the form $ax^2+bx+c=0$ :

$\begin{align} \simplify{{a1}x^2+{a2}x+{a3}}&=\var{a4}\\ \simplify{{a1}x^2+{a2}x+{a3-a4}}&=0\text{.} \end{align}$

Then we can read off values for $a$ , $b$ and $c$ :

$\begin{align} a&=\var{a1}\\ b&=\var{a2}\\ c&=\var{a3-a4} \text{.} \end{align}$

We can now substitute these values into the quadratic formula:

$x = {\frac {-\var{a2}\pm\sqrt{\var{a2}^2-4\times \var{a1}\times \var{a3-a4}}}{2\times\var{a1}}}\text{.}$

So the two solutions are

$\begin{align} x_1&=\var{dpformat(x1,2)}\\ x_2&=\var{dpformat(x2,2)}\text{.} \end{align}$

c)

We first rearrange our equation into the form $ax^2+bx+c=0$ :

$\begin{align} \simplify{{b1}x^2+{b2}x+{b3}}&=0=\var{b4}x\\ \simplify{{b1}x^2+{b2-b4}x+{b3}}&=0\text{.} \end{align}$

We can then read off the values for $a, b$ and $c$ , which are

$\begin{align} a&=\var{b1}\text{,}\\ b&=\var{b2-b4}\text{,}\\ c&=\var{b3}\text{.} \end{align}$

Substituting these values into the quadratic formula,

$x = {\frac {-\var{b2-b4}\pm\sqrt{\var{b2-b4}^2-4\times \var{b1}\times \var{b3}}}{2\times\var{b1}}},$

we obtain solutions

$\begin{align} x_1&=\var{dpformat(p1,2)}\text{,}\\ x_2&=\var{dpformat(p2,2)}\text{.} \end{align}$

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Emma's copy of Using the Quadratic Formula to Solve Equations of the Form $ax^2 +bx+c=0$

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Taxonomy: mathcentre

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History

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Checkpoint description

Emma Cliffe 7 years, 7 months ago

Error in variable testing condition

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Warning

Generated value: `number`

Parts

Gap-fill

a)

b)

c)

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Emma's copy of Using the Quadratic Formula to Solve Equations of the Form ax2+bx+c=0ax^2 +bx+c=0

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Emma Cliffe 7 years, 7 months ago

Error in variable testing condition

Error

Warning

Generated value: number

Parts

Gap-fill

a)

b)

c)

Emma's copy of Using the Quadratic Formula to Solve Equations of the Form $ax^2 +bx+c=0$

Generated value: `number`