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
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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 Quadratic Equations
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 Quadratic Equations
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 Quadratic Equations
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 Quadratic Equations
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
Solving Quadratic Equations
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
Solving Quadratic 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
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 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 Quadratic Equations
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 Quadratic Equations
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 Quadratic Equations
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 Quadratic Equations
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
Solving Quadratic Equations