Use the quadratic formula to solve an equation in terms of an unknown variable

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Xiaodan Leng 5 years, 9 months ago

Created this as a copy of Use the quadratic formula to solve an equation in terms of an unknown variable.

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

The coefficients of the following equation involve the unknown value $k$ . We can use the quadratic formula to find expressions for the values of $x$ in terms of $k$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

b	number	29
c	number	100
n2	number	9
b_2	number	100
c_2	number	22.75
b_3	number	10

			Name	Type	Generated Value

a1	integer	3
a2	integer	24
a3	integer	-24
a4	integer	11
b1	integer	7
b2	integer	23
b3	integer	-5
b4	integer	1
x1	number	-9.2599112794
p1	number	-3.3557137138
p2	number	0.212856571
x2	number	1.2599112794
a	integer	15
m	integer	10

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2c^0.5+a1^2

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Generated value: `number`

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1. Gap Gap 0. Mathematical expression
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Solve the equation $\simplify {x^2+{b_3}*k*x+{c_2}k^2=0}$ . Give your answer in terms of $k$ . Assuming $k$ is positive, enter the lowest root first.

$x_1=$ Gap 0.

$x_2=$ Gap 1.

Advice

The quadratic formula is

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

We can list our values for $a, b$ and $c$ .

$\begin{align} a&=1\\ b&=\var{b_3}k\\ c&=\var{c_2}k^2 \end{align}$

Then by substituting them into the quadratic formula, we obtain

$x=\frac {-\var{b_3}k\pm\sqrt{\var{b_3}^2k^2-4\times \var{c_2}k^2}}{2}$

We can then simplify this equation to

$\begin{align} x&=\frac {-\var{b_3}k\pm k\sqrt{\var{b_3}^2-\var{4c_2}}}{2}\\ \end{align}$

$\begin{align} &=k\left(\frac{-\var{b_3}}{2}\pm \frac {\sqrt{\var{b_3}^2-\var{4c_2}}}{2}\right)\\ \end{align}$

$\begin{align} &=k\left(-\frac{\var{b_3}}{2} \pm\frac{\sqrt{\var{(b_3^2-4c_2)}}}{2}\right)\\ \end{align}$

This means our possible values for $x$ in terms of $k$ are,

$\begin{align} x_1 &= \left( \simplify[all,!noleadingminus,!collectnumbers,!simplifyfractions]{-{b_3}/2 - {sqrt(b_3^2-4c_2)}/2} \right) k = \var[fractionnumbers]{-b_3/2 - sqrt(b_3^2-4*c_2)/2}k \\ x_2 &= \left( \simplify[all,!noleadingminus,!collectnumbers,!simplifyfractions]{-{b_3}/2 + {sqrt(b_3^2-4c_2)}/2} \right) k = \var[fractionnumbers]{-b_3/2 + sqrt(b_3^2-4*c_2)/2}k \end{align}$

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Use the quadratic formula to solve an equation in terms of an unknown variable

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Xiaodan Leng 5 years, 9 months ago

Error in variable testing condition

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Generated value: `number`

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Use the quadratic formula to solve an equation in terms of an unknown variable

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Xiaodan Leng 5 years, 9 months ago

Error in variable testing condition

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Warning

Generated value: number

Parts

Gap-fill

Generated value: `number`