Quadratics: Using the Quadratic Formula

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Ben McGovern

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3 years, 8 months ago

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Ben McGovern 2 years, 6 months ago

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Ben McGovern 3 years, 8 months ago

Gave some feedback: Ready to use

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Ben McGovern 4 years, 9 months ago

Created this as a copy of Surds: Simplifying Expressions 7 - Rationalising the Denominator.

Name	Status	Author	Last Modified
Surds: Simplifying Expressions 1a	Ready to use	Ben McGovern	15/08/2023 15:04
Surds: Simplifying Expressions 1b	Ready to use	Ben McGovern	15/08/2023 15:04
Surds: Simplifying Expressions 2b	Ready to use	Ben McGovern	15/08/2023 15:49
Surds: Simplifying Expressions 2a	Ready to use	Ben McGovern	15/08/2023 15:49
Surds: Simplifying Expressions 3a	Ready to use	Ben McGovern	15/08/2023 15:31
Surds: Simplifying Expressions 3b	Ready to use	Ben McGovern	15/08/2023 15:14
Surds: Simplifying Expressions 4a	Ready to use	Ben McGovern	15/08/2023 15:29
Surds: Simplifying Expressions 4b	Ready to use	Ben McGovern	15/08/2023 15:26
Surds: Simplifying Expressions 4c	Ready to use	Ben McGovern	15/08/2023 15:50
Surds: Simplifying Expressions 5a - Rationalising the Denominator	Ready to use	Ben McGovern	15/08/2023 15:55
Surds: Simplifying Expressions 5b - Rationalising the Denominator	Ready to use	Ben McGovern	15/08/2023 16:07
Surds: Simplifying Expressions 6 - Rationalising the Denominator	Ready to use	Ben McGovern	21/09/2022 16:31
Surds: Simplifying Expressions 7 - Rationalising the Denominator	Ready to use	Ben McGovern	21/09/2022 16:31
Quadratics: Using the Quadratic Formula	Ready to use	Ben McGovern	14/09/2022 16:14
Quadratics: Factorisation 1	Ready to use	Ben McGovern	14/08/2023 15:01
Quadratics: Factorisation 2	Ready to use	Ben McGovern	14/08/2023 15:14
Quadratics: Completing the Square 1	Ready to use	Ben McGovern	14/08/2023 14:59
Quadratics: Completing the Square 2	Ready to use	Ben McGovern	14/09/2022 16:03
Quadratics: Interpreting the Discriminant	Ready to use	Ben McGovern	14/09/2022 16:08
Quadratics: Factorisation 3 - Difference of 2 squares	Ready to use	Ben McGovern	14/08/2023 15:33
Indices: Simplification 1a	Ready to use	Ben McGovern	26/06/2023 12:12
Indices: Simplification 1b	Ready to use	Ben McGovern	26/06/2023 14:31
Indices: Simplification 1c	Ready to use	Ben McGovern	26/06/2023 14:30
Indices: Simplification 2a	Ready to use	Ben McGovern	26/06/2023 14:33
Indices: Simplification 2b	Ready to use	Ben McGovern	26/06/2023 14:36
Indices: Simplification 3a	Ready to use	Ben McGovern	26/06/2023 14:37
Indices: Simplification 3b	Ready to use	Ben McGovern	26/06/2023 14:40
Indices: Change of Notation 1	Ready to use	Ben McGovern	15/08/2023 11:43
Indices: Change of Notation 2	Ready to use	Ben McGovern	15/08/2023 11:51
Indices: Change of Notation 3	Ready to use	Ben McGovern	15/08/2023 11:54
Solve Quadratics - Factorisation 1	Ready to use	Ruth Hand	14/08/2023 15:44
Quadratics: Solving a Quadratic Equation	Ready to use	Ben McGovern	09/12/2022 16:45
Quadratics: Solving by Factorisation	Ready to use	Ben McGovern	14/09/2022 16:25
AD2 Quadratics - solve	Ready to use	Megan Oliver	09/04/2024 15:13
AD5 Difference of two squares	Ready to use	Megan Oliver	09/04/2024 15:14
AA2 Indices - divide	Ready to use	Andrew Neate	09/04/2024 15:00
AA4 Indices - Fractional 2	Ready to use	Andrew Neate	09/04/2024 15:01
AA3 Indices - Fractional 1	Ready to use	Andrew Neate	09/04/2024 15:01
AA1 Indices - times	Ready to use	Andrew Neate	28/03/2024 13:17
Calculation with indices - times	Ready to use	Will Morgan	26/06/2024 11:53
Calculation with Indices - divide	Ready to use	Will Morgan	26/06/2024 12:07

There are 209 other versions that do you not have access to.

Question statement

Using the quadratic formula, solve the following quadratic equation:

$\simplify{{a}x^2+{b}x+{c}=0}$

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	2
b	integer	-3
c	integer	-1
solx1	number	1.7807764064
solx2	number	-0.2807764064
sol1	number	1.781
sol2	number	-0.281

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xxxxxxxxxx
 
random(1..5)

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Generated value: `integer`

→ Used by:

b
c
solx1
solx2

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Advice

For a quadratic equation of the form

$ax^2+bx+c = 0,$ we can use the quadratic formula to find solutions for x:

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

For the equation

$\simplify{{a}x^2+{b}x+{c}=0},$

the values of the coefficients are

$a=\var{a}, \quad b=\var{b}, \quad c=\var{c}.$

Substituting these values into the quadratic formula:

$\begin{split} x &= \frac{\simplify{-{b}} \pm \sqrt{(\var{b})^2 - (4 \times \var{a} \times \var{c})}}{2 \times \var{a}} \\\\&= \frac{\simplify{-{b}} \pm \simplify[!collectNumbers]{sqrt({b^2}-{4*a*c})}}{\var{2a}} \\\\&= \frac{\simplify{-{b}} \pm \simplify[!collectNumbers]{sqrt({b^2-4*a*c})}}{\var{2a}} \\\\&= \frac{\simplify{-{b}} \pm \simplify{sqrt({b^2-4*a*c})}}{\var{2a}}. \end{split}$

Note the $\pm$ symbol in the quadratic formula. This means that there are $2$ solutions for $x$ : one using $+$ and one using $-$ .

Therefore,

$\begin{split} x_1 &\,= \frac{\simplify{-{b}} + \simplify{sqrt({b^2-4*a*c})}}{\var{2a}} \qquad \text{and} \qquad x_2 &\,= \frac{\simplify{-{b}} - \simplify{sqrt({b^2-4*a*c})}}{\var{2a}}\\\\ &\,= \var{sol1},\quad &\,=\var{sol2}. \end{split}$

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This question is used in the following exams:

Algebra basics by Ruth Hand in MASH Bath: Moodle quizzes and TS.
Quadratic Functions pre A-level by Ruth Hand in MASH Bath: Moodle quizzes and TS.
Quadratics by Ben McGovern in MASH Bath: Moodle quizzes and TS.
Quadratics practice by Ruth Douglas in Ruth's workspace.
All about quadratics by Jinhua Mathias in Maths 3-2020.
Maths 1A Week 2 Online Homework by Maria Aneiros in Maria's workspace.
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Solving Linear and Quadratic Equations Quiz by Ruth Hand in MASH Bath: Moodle quizzes and TS.
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Scientific Maths Training Space by Ruth Hand in MASH Bath: Moodle quizzes and TS.
Mixed practice on quadratic equations by Jinhua Mathias in Maths 3-2020.
Summer Quiz 1 by Kate Vincent in Kate's workspace.

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Ben McGovern 2 years, 6 months ago

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Quadratics: Using the Quadratic Formula

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Ben McGovern 2 years, 6 months ago

Ben McGovern 3 years, 8 months ago

Ben McGovern 4 years, 9 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`