5: Solving Equations © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.

5: Solving Equations5: Solving Equations

© Christine Crisp

““Teach A Level Maths”Teach A Level Maths”

Vol. 1: AS Core Vol. 1: AS Core ModulesModules

Solving Equations

Module C1

"Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"

The value of the expression can be found for any value of the unknown, x

e.g. is an expression123 2 xx

71412

e.g. is a quadratic equation0123 2 xxThese equations can be solved. There is one value satisfying the 1st equation and two values which satisfy the 2nd equation.

e.g. is an identity))(( 113123 2 xxxx

An identity is true for all values of the unknown.

1232 2 xxxe.g.

e.g. is a linear equation012 x

( Identities are sometimes written with instead of = )

Expressions, Equations and Identities

Solving Equations

Solving Linear Equations Collect the terms containing the

unknown on one side of the equation and the constants on the other

e.g. 743 xx

473 xx

112 x

2

11 x

Linear equations only have constants and x-terms without powers.

Solving Equations

02 xx

e.g. 1 xx 2

01 )(xx

0x 01 x1x

or

Get zero on one side

( Common factor )

Two factors multiplied together = 0,

so one must be zero.

Try to factorise

Do NOT cancel x as a solution will then be lost.

Solving Quadratic Equations

Solving Equations

e.g. 2 672 xx

016 ))(( xx

Zero on one side

( Trinomial )

Two factors multiplied together = 0, so one factor must equal zero.

Try to factorise

0672 xx

1x 6x or

2316or

01 x 06 x or


Solving Equations

e.g. 3 0872 xx

Multiply by -1

Trinomial

Two factors multiplied together = 0, so one factor must be zero.

Try to factorise

0872 xx

1x8x or

2418or

018 ))(( xx


Solving Equations

42 x

e.g. 4 xx

4

In this example there is no linear term. Instead of getting 0 on the r.h.s. we can square root directly.

Multiply by x

2x

N.B.


Solving Equations

Exercises Solve the following quadratic equations

122

xx

0652 xx

052 xx

052 x

xx

9

0252 2 xx

1.

2.

3.

4.

5.

6.

0)1)(6( xx 1,6 x

0)5( xx 50, x

52 x 55, x

0122

xx0)3)(4( xx 3,4 x

92

x 3 x

0)2)(12( xx 2,21 x

Solutions

Solving Equations

If a quadratic equation is written as

02 cbxaxthen if is a perfect square, the quadratic will factorise

acb 42

[ ]))(( 13232 2 xxxx

))(()( 32414 22 acb

The quadratic factorises!

e.g. 2 0352 xx

))(( 3142542 acb

The quadratic does not factorise!

32 2 xxe.g. 1

25241

351 cba ,,

13

312 cba ,,

A useful tip:

Solving Equations

e.g. 5 0142 xx

This quadratic doesn’t factorise so complete the square 0142 2 )(x

032 2 )(x

32 2 )(x

To solve for x, we need to square root, so we isolate the squared term on the left of the equal sign (l.h.s.)

Square rooting

32 x

32 x

These answers are exact but can be given as approximate decimals.

N.B. 2 Solutions!


Solving Equations

The method used in the last example can be generalised to give us a formula which is easier to use when the coefficient of is not 1 2x


The formula will be proved but you don’t need to know the proof.

However, you must memorise the result.

Solving Equations

Consider

02 cbxax

022

22

a

c

a

b

a

bx

Complete the square:

Divide by a:

02 a

cx

a

bx

a

c

a

b

a

bx

2

22

42

a

c

a

b

a

bx

2

2

42

2

2

4

4

2 a

acb

a

bx

a

acbbx

2

42

Proof of the Quadratic Formula

Solving Equations

453 cba ,,

e.g. 6 Solve the equation 0453 2 xx

6

43455 2 ))(()( x

6

735 x

6

48255 x

a

acbbx

2

42 Solution

:

6

735 xo

r


Zero on one side

Try to factorise

If there are no factors, complete the square ( if a = 1 ) or use the formula

• Common Factors

• Trinomial factors

If there are factors, factorise and solve

a

acbbx

2

42

EXCEPTION: If there is no ‘x’ term write the equation as and square root.

cx 2

Solving Quadratic Equations - SUMMARY

Solving Equations

Exercises Use the most efficient method to solve the following quadratic equations:0642 xx

0152 2 xx

01522 xx

1.

2.

3.

064)2( 2 x

0)3)(5( xx

a

acbbx

2

42

1,5,2 cba

4

8255 x

Solution:

Complete the Square 10)2( 2 x

102 x 102 x

Solution:

Use the formula.

4

335

3,5 xSolution: Factorise

Solving Equations

Solving Equations

The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

Solving Equations

The value of the expression can be found for any value of the unknown, x

e.g. is an expression123 2 xx

71412

e.g. is a quadratic equation0123 2 xxThese equations can be solved. There is one value satisfying the 1st equation and two values which satisfy the 2nd equation.

e.g. is an identity))(( 113123 2 xxxx

An identity is true for all values of the unknown.

1232 2 xxxe.g.

e.g. is a linear equation012 x

( Identities can be written as but only for emphasis.)

Expressions, Equations and Identities

Solving EquationsSolving Linear

Equations

Collect the terms containing the unknown on one side of the equation and the constants on the othere.g. 743 xx

473 xx

112 x

2

11 x

Linear equations only have constants and x-terms without powers.

Solving Equations

Zero on one side

Try to factorise

If there are no factors, complete the square ( if a = 1 ) or use the formula

• Common Factors

• Trinomial factors

If there are factors, factorise and solve

a

acbbx

2

42

EXCEPTION: If there is no ‘x’ term write the equation as and square root.

cx 2

Solving Quadratic Equations - SUMMARY

Solving Equations

02 xx

e.g. 1 xx 2

01 )(xx

0x 01 x1x

or

Get zero on one side

( Common factor )

Two factors multiplied together = 0,so one must be zero.

Try to factorise

Do NOT cancel x as a solution will then be lost.


Solving Equations

e.g. 2 672 xx

016 ))(( xx

Zero on one side ( Trinomial )

Two factors multiplied together = 0, so one factor must equal zero.

Try to factorise

0672 xx

1x 6x or

2316or

01 x 06 x or


Solving Equations

e.g. 3 0872 xx

Multiply by -1

Trinomial

Two factors multiplied together = 0, so one factor must be zero.

Try to factorise

0872 xx

1x8x or

2418or

018 ))(( xx


Solving Equations

42 x

e.g. 4 xx

4

In this example there is no linear term. Instead of getting 0 on the r.h.s. we can square root directly.

Multiply by x

2x

N.B.


Solving Equations

e.g. 5 0142 xx

This quadratic doesn’t factorise so complete the square 0142 2 )(x

032 2 )(x

32 2 )(x

To solve for x, we need to square root, so we isolate the squared term on the left of the equal sign (l.h.s.)

Square rooting

32 x

32 x

These answers are exact but can be given as approximate decimals.

N.B. 2 Solutions!


Solving Equations

453 cba ,,

e.g. 6 Solve the equation 0453 2 xx

6

43455 2 ))(()( x

6

735 x

6

48255 x

a

acbbx

2

42 Solution

:

6

735 xo

r


5: Solving Equations © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.

Documents

Transcript of 5: Solving Equations © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.