CAMBRIDGE 'A' LEVELS 1

2010-3-30 P1/1:QUADRATICS 1

QUADRATICS

P1/1/1: Quadratic expressions of the form

ax2 + bx + c and their graphs

P1/1/2: Solving quadratic equation in one unknown

P1/1/3: Nature of roots of quadratic expression

P1/1/4: Simultaneous equations of which one is

linear and one is quadratic

P1/1/5: Linear inequalities and quadratic inequalities

P1/1/6: Summary of lessonPrepared by

Tan Bee Hong

P1/1/1

Quadratic expressions of the form

ax2 + bx + c and their graphs

2010-3-30 P1/1:QUADRATICS 3

Learning Outcome

Students should be able to:

• carry out the process of completing the square for a

quadratic polynomial.

• locate the vertex of the quadratics graphs from the

completed form.

• sketch the quadratic graph

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Quadratics expression

nts).(coefficie constants are c and b 0),a(a where

2

≠

++ cbxax

The graph is a parabola.

If a > 0 or

If a < 0

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Completed square form

as written becan 2110)( 2 +−= xxxf

formfactor )3)(7()( −−= xxxf x-intercept?

form square completed 4)5()( 2 −−= xxfVertex?

Range of f(x)?

Graph

2010-3-30 P1/1:QUADRATICS 6

Completing the square

2

2

2

22

2

4

1

2

1

4

1

2

1

bbxbxx

bbxxbx

−

+=+⇒

++=

+

form square completed expression quadratic 2 ⇒++ cbxx

pointkey

In general,

Both sides plus c: cbbxcbxx +−

+=++ 2

2

2

4

1

2

1

CAMBRIDGE 'A' LEVELS 2

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Example 1:

50142 ++ xx

Graph

Express in completed square form.

Locate the vertex and the axis of symmetry of the

quadratic graph. Find the least or greatest value of

the expression, and the value of x for which this

occurs.

2010-3-30 P1/1:QUADRATICS 8

Example 2:

51222 −+ xx

Graph

Express in completed square form.

Locate the vertex and the axis of symmetry of the

quadratic graph. Find the least or greatest value of

the expression, and the value of x for which this

occurs.

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Example 3:

2373 xx −−

Graph

Express in completed square form.

Locate the vertex and the axis of symmetry of the

quadratic graph. Find the least or greatest value of

the expression, and the value of x for which this

occurs.

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Example 4:

Express in completed square form, and

use your result to find the factors of .

612 2 −+ xx

Graph

612 2 −+ xx

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Example 5:

Find the range of the function

)2)(1()( −−= xxxf

Graph

2010-3-30 P1/1:QUADRATICS 12

Practice Exercise

Pure Mathematics 1 Hugh Neil & Douglas Quadling (2002)

Exercise 4A (Page 54)

Q3(d), Q5(e), Q6(e)(f), Q8(c)(f), Q9(c)

CAMBRIDGE 'A' LEVELS 3

P1/1/2

Solving quadratic equation in

one unknown

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Learning Outcome

Students should be able to:

• use an appropriate method to solve a given quadratic

equation.

• solve equations which can be reduced to quadratic

equations.

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Solving Quadratic equation in one unknown

Solving quadratic equations by:

(i) Factorization

Example 6:

0432 =−+ xx

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Solving Quadratic equation in one unknown

Solving quadratic equations by:

(ii) Completing the square method

Example 7:

03722 =++ xx

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Solving Quadratic equation in one unknownSolving quadratic equations by:

(iii) Quadratic formula

a

acbbx

acbxax

2

4

is ,0 where,0 ofsolution The

2

2

−±−=

≠=++

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Example 8:

Use the quadratic formula to solve the following

equations. Leave your answers in surd form. If

there is no solution, say so.

0432 )( 2 =−− xxa

0432 )( 2 =+− xxb

CAMBRIDGE 'A' LEVELS 4

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Equation which reduce to quadratic equations

Example 9:

Solve the equation 04524 =+− xx

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Equation which reduce to quadratic equations

Example 10:

Solve the equation xx −= 6

equation theof sidesboth squaringby (b)

for stand lettingby (a) xy

2010-3-30 P1/1:QUADRATICS 21

Practice Exercise

Pure Mathematics 1 Hugh Neil & Douglas Quadling (2002)

Exercise 4B (Page 58)

Q1(e)(g)(i)

Exercise 4C (Page 61)

Q4(d)(f), Q5(f)(l), Q6(d)(e)

P1/1/3

Nature of roots of

quadratic expression

2010-3-30 P1/1:QUADRATICS 23

Learning Outcome

Students should be able to:

• evaluate the discriminant of a quadratic polynomial.

• use the discriminant to determine the nature of the roots.

• relate the nature of roots to the quadratic graph.

2010-3-30 P1/1:QUADRATICS 24

Nature of roots of quadratic expression

acb 4nt discrimina The2 −

a

acbbx

cbxax

2

4

0

2

2

−±−=⇒

=++

(ii) If b2 – 4ac > 0, the equation ax2 + bx + c = 0 will have

two roots.

(iii) If b2 – 4ac < 0, there will be no roots.

(iv) If b2 – 4ac = 0, there is one root only or a repeated root.

CAMBRIDGE 'A' LEVELS 5

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Example 11:

What can you deduce from the values of discriminants of

the quadratics in the following equations?

0372 )(2 =+− xxa

043 )( 2 =+− xxb

012 )( 2 =++ xxc

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Example 12:

The equation 3x2 + 5x – k = 0 has two real roots. What can

you deduce about the value of the constant k?

The equation 3x2 + 5x – k = 0 has two distinct real roots.

What can you deduce about the value of the constant k?

2010-3-30 P1/1:QUADRATICS 27

Example 13:

The equation x2 + kx + 9 = 0 has no root. Deduce as much

as you can about the values of k?

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Example 14:

The equation -3 + kx - 2x2 = 0 has a repeated root. Find the

values of k. (exact fractions or surds)

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Practice Exercise

Pure Mathematics 1 Hugh Neil & Douglas Quadling (2002)

Exercise 4B (Page 58)

Q4(e)(f)(g), Q5(e)(h)(i)

P1/1/4

Simultaneous equations of which

one is linear and one is quadratic

CAMBRIDGE 'A' LEVELS 6

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Learning Outcome

Students should be able to:

• solve by substitution a pair of simultaneous equations

of which one is linear and one is quadratic.

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Simultaneous equations of which one is linear and

one is quadratic

Example 15:

Solve the simultaneous equations

3 ,832 −=−−= xyxxy

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Simultaneous equations of which one is linear and

one is quadratic

Example 16:

Solve the simultaneous equations

122 ,2734 22 −=−=−+ xyyxyx

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Simultaneous equations of which one is linear and

one is quadratic

Example 17:

At how many points does the line 3y – x = 15 meet the

curve 4x2 + 9y2 = 36.

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Practice Exercise

Pure Mathematics 1 Hugh Neil & Douglas Quadling (2002)

Exercise 4C (Page 61)

Q2(h), Q3(d)

P1/1/5

Linear inequalities and

quadratic inequalities

CAMBRIDGE 'A' LEVELS 7

2010-3-30 P1/1:QUADRATICS 37

Learning Outcome

Students should be able to:

• solve linear inequalities

• solve quadratic inequalities

2010-3-30 P1/1:QUADRATICS 38

Linear inequalities and quadratic inequalities

} esinequaliti Strictabba <⇔>

} esinequaliti Weakabba ≤⇔≥

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Algebraic method

axaxax

axaax

a

≥−≤⇔≥

≤≤−⇔≤

>

or

equivalent are statements e then thes,0 If

22

22

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Example 18:

Solve the linear inequalities.

( ) ( ) 1032183

1 )(

123 )(

573 )(

>−−+

−≥−

−>+

xxiii

xii

xi

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Example 19:Solve the quadratic inequalities by:

(a) Graphical method

(b) Tabular method

( )( )( )( ) 04343 )(

014 )(

>+−

≥+−

xxii

xxi

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Example 20:Solve the quadratic inequalities by using the algebraic

method.

053 )(

038 )(

2

2

>−+

>−−

xxii

xxi

CAMBRIDGE 'A' LEVELS 8

2010-3-30 P1/1:QUADRATICS 43

Practice Exercise

Pure Mathematics 1 Hugh Neil & Douglas Quadling (2002)

Exercise 5A (Page 68)

Q3(g), Q4(h), 5(i), 6(g)

Exercise 5B (Page 71)

Q1(i)(k), Q3(b)(c)(d)