Great Issues of Our Time: The Quadratic Formula William McCallum Tucson Teachers’ Circle, March 26, 2008 William McCallum Great Issues of Our Time: The Quadratic Formula
Great Issues of Our Time:The Quadratic Formula
William McCallum
Tucson Teachers’ Circle, March 26, 2008
William McCallum Great Issues of Our Time: The Quadratic Formula
A proposal to eliminate quadratic equations
21 April 2003
Terry Bladen, president of the National Association ofSchoolmasters Union of Women Teachers:
“pupils should be numerate . . . butnumeracy can be divorced frommathematics. . . . How often do the majorityof people need or use mathematicalconcepts once they have left school?”
[He advocated] allowing them to drop advanced concepts suchas quadratic equations and trigonometry at the age of 14.
William McCallum Great Issues of Our Time: The Quadratic Formula
A proposal to eliminate quadratic equations
21 April 2003
Terry Bladen, president of the National Association ofSchoolmasters Union of Women Teachers:
“pupils should be numerate . . . butnumeracy can be divorced frommathematics. . . . How often do the majorityof people need or use mathematicalconcepts once they have left school?”
[He advocated] allowing them to drop advanced concepts suchas quadratic equations and trigonometry at the age of 14.
William McCallum Great Issues of Our Time: The Quadratic Formula
A proposal to eliminate quadratic equations
21 April 2003
Terry Bladen, president of the National Association ofSchoolmasters Union of Women Teachers:
“pupils should be numerate . . . butnumeracy can be divorced frommathematics. . . . How often do the majorityof people need or use mathematicalconcepts once they have left school?”
[He advocated] allowing them to drop advanced concepts suchas quadratic equations and trigonometry at the age of 14.
William McCallum Great Issues of Our Time: The Quadratic Formula
The proposal is debated in parliament
26 June 2003Tony McWalter, Labour MP
“A quadratic equation is not like a bleakroom, devoid of furniture, in which one isasked to squat. It is a door to a room full ofthe unparalleled riches of human intellectualachievement. If you do not go through thatdoor . . . much that passes for humanwisdom will be forever denied you.”
“Hear, hear”—Eleanor Laing, Conservative MP
“Oh dear. I would like to have support fromelsewhere as well.”
William McCallum Great Issues of Our Time: The Quadratic Formula
The proposal is debated in parliament
26 June 2003Tony McWalter, Labour MP
“A quadratic equation is not like a bleakroom, devoid of furniture, in which one isasked to squat. It is a door to a room full ofthe unparalleled riches of human intellectualachievement. If you do not go through thatdoor . . . much that passes for humanwisdom will be forever denied you.”
“Hear, hear”—Eleanor Laing, Conservative MP
“Oh dear. I would like to have support fromelsewhere as well.”
William McCallum Great Issues of Our Time: The Quadratic Formula
The proposal is debated in parliament
26 June 2003Tony McWalter, Labour MP
“A quadratic equation is not like a bleakroom, devoid of furniture, in which one isasked to squat. It is a door to a room full ofthe unparalleled riches of human intellectualachievement. If you do not go through thatdoor . . . much that passes for humanwisdom will be forever denied you.”
“Hear, hear”—Eleanor Laing, Conservative MP
“Oh dear. I would like to have support fromelsewhere as well.”
William McCallum Great Issues of Our Time: The Quadratic Formula
The proposal is debated in parliament
26 June 2003Tony McWalter, Labour MP
“A quadratic equation is not like a bleakroom, devoid of furniture, in which one isasked to squat. It is a door to a room full ofthe unparalleled riches of human intellectualachievement. If you do not go through thatdoor . . . much that passes for humanwisdom will be forever denied you.”
“Hear, hear”—Eleanor Laing, Conservative MP
“Oh dear. I would like to have support fromelsewhere as well.”
William McCallum Great Issues of Our Time: The Quadratic Formula
The proposal is debated in parliament
26 June 2003Tony McWalter, Labour MP
“A quadratic equation is not like a bleakroom, devoid of furniture, in which one isasked to squat. It is a door to a room full ofthe unparalleled riches of human intellectualachievement. If you do not go through thatdoor . . . much that passes for humanwisdom will be forever denied you.”
“Hear, hear”—Eleanor Laing, Conservative MP
“Oh dear. I would like to have support fromelsewhere as well.”
William McCallum Great Issues of Our Time: The Quadratic Formula
The minister replies
26 June 2003Alan Johnson, Minister
“ In preparing for this debate, the DFESconducted a straw poll involving a 16-year-oldwho had just sat maths GCSE, a head ofmaths and an experienced chemical engineer.”
The 16-year-old thought that quadratic equations werelogical and fairly straightforward because ‘you substitutestuff into a formula’. . . .The head of maths said that quadratic equations formedan important step in students’ ability to solve equations, . . .The engineer said that he did not use quadratic equationsnow, but had in the past . . . ”
William McCallum Great Issues of Our Time: The Quadratic Formula
The minister replies
26 June 2003Alan Johnson, Minister
“ In preparing for this debate, the DFESconducted a straw poll involving a 16-year-oldwho had just sat maths GCSE, a head ofmaths and an experienced chemical engineer.”
The 16-year-old thought that quadratic equations werelogical and fairly straightforward because ‘you substitutestuff into a formula’. . . .The head of maths said that quadratic equations formedan important step in students’ ability to solve equations, . . .The engineer said that he did not use quadratic equationsnow, but had in the past . . . ”
William McCallum Great Issues of Our Time: The Quadratic Formula
The minister replies
26 June 2003Alan Johnson, Minister
“ In preparing for this debate, the DFESconducted a straw poll involving a 16-year-oldwho had just sat maths GCSE, a head ofmaths and an experienced chemical engineer.”
The 16-year-old thought that quadratic equations werelogical and fairly straightforward because ‘you substitutestuff into a formula’. . . .
The head of maths said that quadratic equations formedan important step in students’ ability to solve equations, . . .The engineer said that he did not use quadratic equationsnow, but had in the past . . . ”
William McCallum Great Issues of Our Time: The Quadratic Formula
The minister replies
26 June 2003Alan Johnson, Minister
“ In preparing for this debate, the DFESconducted a straw poll involving a 16-year-oldwho had just sat maths GCSE, a head ofmaths and an experienced chemical engineer.”
The 16-year-old thought that quadratic equations werelogical and fairly straightforward because ‘you substitutestuff into a formula’. . . .The head of maths said that quadratic equations formedan important step in students’ ability to solve equations, . . .
The engineer said that he did not use quadratic equationsnow, but had in the past . . . ”
William McCallum Great Issues of Our Time: The Quadratic Formula
The minister replies
26 June 2003Alan Johnson, Minister
“ In preparing for this debate, the DFESconducted a straw poll involving a 16-year-oldwho had just sat maths GCSE, a head ofmaths and an experienced chemical engineer.”
The 16-year-old thought that quadratic equations werelogical and fairly straightforward because ‘you substitutestuff into a formula’. . . .The head of maths said that quadratic equations formedan important step in students’ ability to solve equations, . . .The engineer said that he did not use quadratic equationsnow, but had in the past . . . ”
William McCallum Great Issues of Our Time: The Quadratic Formula
Al-Khwarizmi, Hisab al-jabr w’al-muqabala, 830
. . . what is the square which combined with ten ofits roots will give a sum total of 39? The mannerof solving this type of equation is to take one-halfof the roots just mentioned. . . . Therefore take 5,which multiplied by itself gives 25, an amountwhich you add to 39 giving 64. Having taken thenthe square root of this which is 8, subtract from ithalf the roots, 5 leaving 3. Exercise
William McCallum Great Issues of Our Time: The Quadratic Formula
Al-Khwarizmi, Hisab al-jabr w’al-muqabala, 830
. . . what is the square which combined with ten ofits roots will give a sum total of 39? The mannerof solving this type of equation is to take one-halfof the roots just mentioned. . . . Therefore take 5,which multiplied by itself gives 25, an amountwhich you add to 39 giving 64. Having taken thenthe square root of this which is 8, subtract from ithalf the roots, 5 leaving 3.
ExerciseWhat equation is al-Khwarizmi talking about here?
Answer Skip
William McCallum Great Issues of Our Time: The Quadratic Formula
Al-Khwarizmi, Hisab al-jabr w’al-muqabala, 830
. . . what is the square which combined with ten ofits roots will give a sum total of 39? The mannerof solving this type of equation is to take one-halfof the roots just mentioned. . . . Therefore take 5,which multiplied by itself gives 25, an amountwhich you add to 39 giving 64. Having taken thenthe square root of this which is 8, subtract from ithalf the roots, 5 leaving 3.
x2 + 10x = 39
x2 + 10x + 25 = 39 + 25 = 64x + 5 = 8
x = 3
William McCallum Great Issues of Our Time: The Quadratic Formula
Al-Khwarizmi, Hisab al-jabr w’al-muqabala, 830
. . . what is the square which combined with ten ofits roots will give a sum total of 39? The mannerof solving this type of equation is to take one-halfof the roots just mentioned. . . . Therefore take 5,which multiplied by itself gives 25, an amountwhich you add to 39 giving 64. Having taken thenthe square root of this which is 8, subtract from ithalf the roots, 5 leaving 3.
x2 + 10x = 39
x2 + 10x + 25 = 39 + 25 = 64x + 5 = 8
x = 3
William McCallum Great Issues of Our Time: The Quadratic Formula
Al-Khwarizmi, Hisab al-jabr w’al-muqabala, 830
. . . what is the square which combined with ten ofits roots will give a sum total of 39? The mannerof solving this type of equation is to take one-halfof the roots just mentioned. . . . Therefore take 5,which multiplied by itself gives 25, an amountwhich you add to 39 giving 64. Having taken thenthe square root of this which is 8, subtract from ithalf the roots, 5 leaving 3.
x2 + 10x = 39
x2 + 10x + 25 = 39 + 25 = 64
x + 5 = 8x = 3
William McCallum Great Issues of Our Time: The Quadratic Formula
Al-Khwarizmi, Hisab al-jabr w’al-muqabala, 830
. . . what is the square which combined with ten ofits roots will give a sum total of 39? The mannerof solving this type of equation is to take one-halfof the roots just mentioned. . . . Therefore take 5,which multiplied by itself gives 25, an amountwhich you add to 39 giving 64. Having taken thenthe square root of this which is 8, subtract from ithalf the roots, 5 leaving 3.
x2 + 10x = 39
x2 + 10x + 25 = 39 + 25 = 64
x + 5 = 8x = 3
William McCallum Great Issues of Our Time: The Quadratic Formula
Al-Khwarizmi, Hisab al-jabr w’al-muqabala, 830
. . . what is the square which combined with ten ofits roots will give a sum total of 39? The mannerof solving this type of equation is to take one-halfof the roots just mentioned. . . . Therefore take 5,which multiplied by itself gives 25, an amountwhich you add to 39 giving 64. Having taken thenthe square root of this which is 8, subtract from ithalf the roots, 5 leaving 3.
x2 + 10x = 39
x2 + 10x + 25 = 39 + 25 = 64x + 5 = 8
x = 3
William McCallum Great Issues of Our Time: The Quadratic Formula
Al-Khwarizmi’s geometric proof of his method
x2 + 10x = 39x2 + 10x + 25 = 39 + 25 = 64
x + 5 = 8x = 3
x25x
5x25
! x "! 5"#
x
$#
5$
William McCallum Great Issues of Our Time: The Quadratic Formula
Al-Khwarizmi’s geometric proof of his method
x2 + 10x = 39x2 + 10x + 25 = 39 + 25 = 64
x + 5 = 8x = 3
x25x
5x25
! x "! 5"#
x
$#
5$
William McCallum Great Issues of Our Time: The Quadratic Formula
Al-Khwarizmi’s geometric proof of his method
x2 + 10x = 39x2 + 10x + 25 = 39 + 25 = 64
x + 5 = 8x = 3
x25x
5x25
! x "! 5"#
x
$#
5$
William McCallum Great Issues of Our Time: The Quadratic Formula
Al-Khwarizmi’s geometric proof of his method
x2 + 10x = 39x2 + 10x + 25 = 39 + 25 = 64
x + 5 = 8x = 3
x25x
5x25
! x "! 5"#
x
$#
5$
William McCallum Great Issues of Our Time: The Quadratic Formula
Al-Khwarizmi’s geometric proof of his method
x2 + 10x = 39x2 + 10x + 25 = 39 + 25 = 64
x + 5 = 8x = 3
Exercise
x25x
5x25
! x "! 5"#
x
$#
5$
William McCallum Great Issues of Our Time: The Quadratic Formula
Al-Khwarizmi’s geometric proof of his method
x2 + 10x = 39x2 + 10x + 25 = 39 + 25 = 64
x + 5 = 8x = 3
x25x
5x25
! x "! 5"#
x
$#
5$
ExerciseDraw a diagram that illustrates the solution of the equation
x2 = 39 + 10x .
Answer Skip
William McCallum Great Issues of Our Time: The Quadratic Formula
Al-Khwarizmi’s geometric proof of his method
x2 + 10x = 39x2 + 10x + 25 = 39 + 25 = 64
x + 5 = 8x = 3
x25x
5x25
! x "! 5"#
x
$#
5$
x2 = 39 + 10x
x2 + 25 = 64 + 10xx − 5 = 8
x = 13
! x "
x2
William McCallum Great Issues of Our Time: The Quadratic Formula
Al-Khwarizmi’s geometric proof of his method
x2 + 10x = 39x2 + 10x + 25 = 39 + 25 = 64
x + 5 = 8x = 3
x25x
5x25
! x "! 5"#
x
$#
5$
x2 = 39 + 10xx2 + 25 = 64 + 10x
x − 5 = 8x = 13
There is a better way.
(x − 5)2
5x
5x
! x "#
x − 5
$#
5$
William McCallum Great Issues of Our Time: The Quadratic Formula
Al-Khwarizmi’s geometric proof of his method
x2 + 10x = 39x2 + 10x + 25 = 39 + 25 = 64
x + 5 = 8x = 3
x25x
5x25
! x "! 5"#
x
$#
5$
x2 = 39 + 10xx2 + 25 = 64 + 10x
x − 5 = 8x = 13
There is a better way.
(x − 5)2
5x
5x
! x "#
x − 5
$#
5$
William McCallum Great Issues of Our Time: The Quadratic Formula
Al-Khwarizmi’s geometric proof of his method
x2 + 10x = 39x2 + 10x + 25 = 39 + 25 = 64
x + 5 = 8x = 3
x25x
5x25
! x "! 5"#
x
$#
5$
x2 = 39 + 10xx2 + 25 = 64 + 10x
x − 5 = 8x = 13
There is a better way.
(x − 5)2
5x
5x
! x "#
x − 5
$#
5$
William McCallum Great Issues of Our Time: The Quadratic Formula
Al-Khwarizmi’s geometric proof of his method
x2 + 10x = 39x2 + 10x + 25 = 39 + 25 = 64
x + 5 = 8x = 3
x25x
5x25
! x "! 5"#
x
$#
5$
x2 = 39 + 10xx2 + 25 = 64 + 10x
x − 5 = 8x = 13
There is a better way.
(x − 5)2
5x
5x
! x "#
x − 5
$#
5$
William McCallum Great Issues of Our Time: The Quadratic Formula
Only positive solutions are allowed
Problems were phrased in terms of areas, weights, money.
Al-Khwarizmi’s method guarantees a unique solution to theequations
x2 + Cx = Nx2 = Cx + N
where N and C are positive numbers.So each equation must have a negative solution as well.Imaginary solutions were far in the future.
William McCallum Great Issues of Our Time: The Quadratic Formula
Only positive solutions are allowed
Problems were phrased in terms of areas, weights, money.Al-Khwarizmi’s method guarantees a unique solution to theequations
x2 + Cx = Nx2 = Cx + N
where N and C are positive numbers.
So each equation must have a negative solution as well.Imaginary solutions were far in the future.
William McCallum Great Issues of Our Time: The Quadratic Formula
Only positive solutions are allowed
Problems were phrased in terms of areas, weights, money.Al-Khwarizmi’s method guarantees a unique solution to theequations
x2 + Cx = Nx2 = Cx + N
where N and C are positive numbers.So each equation must have a negative solution as well.
Imaginary solutions were far in the future.
William McCallum Great Issues of Our Time: The Quadratic Formula
Only positive solutions are allowed
Problems were phrased in terms of areas, weights, money.Al-Khwarizmi’s method guarantees a unique solution to theequations
x2 + Cx = Nx2 = Cx + N
where N and C are positive numbers.So each equation must have a negative solution as well.Imaginary solutions were far in the future.
William McCallum Great Issues of Our Time: The Quadratic Formula
Completing the square the modern way
x2 + bx + c = 0
x2 + bx = − c
x2 + bx +
(b2
)2= − c +
(b2
)2
(x +
b2
)2=
b2 − 4c4
x +b2
= ±√
b2 − 4c2
x =−b ±
√b2 − 4c
2
William McCallum Great Issues of Our Time: The Quadratic Formula
Completing the square the modern way
x2 + bx + c = 0
x2 + bx = − c
x2 + bx +
(b2
)2= − c +
(b2
)2
(x +
b2
)2=
b2 − 4c4
x +b2
= ±√
b2 − 4c2
x =−b ±
√b2 − 4c
2
William McCallum Great Issues of Our Time: The Quadratic Formula
Completing the square the modern way
x2 + bx + c = 0
x2 + bx = − c
x2 + bx +
(b2
)2= − c +
(b2
)2
(x +
b2
)2=
b2 − 4c4
x +b2
= ±√
b2 − 4c2
x =−b ±
√b2 − 4c
2
William McCallum Great Issues of Our Time: The Quadratic Formula
Completing the square the modern way
x2 + bx + c = 0
x2 + bx = − c
x2 + bx +
(b2
)2= − c +
(b2
)2
(x +
b2
)2=
b2 − 4c4
x +b2
= ±√
b2 − 4c2
x =−b ±
√b2 − 4c
2
William McCallum Great Issues of Our Time: The Quadratic Formula
Completing the square the modern way
x2 + bx + c = 0
x2 + bx = − c
x2 + bx +
(b2
)2= − c +
(b2
)2
(x +
b2
)2=
b2 − 4c4
x +b2
= ±√
b2 − 4c2
x =−b ±
√b2 − 4c
2
William McCallum Great Issues of Our Time: The Quadratic Formula
Completing the square the modern way
x2 + bx + c = 0
x2 + bx = − c
x2 + bx +
(b2
)2= − c +
(b2
)2
(x +
b2
)2=
b2 − 4c4
x +b2
= ±√
b2 − 4c2
x =−b ±
√b2 − 4c
2
William McCallum Great Issues of Our Time: The Quadratic Formula
Completing the square the modern way
ax2 + bx + c = 0
x2 +ba
x =−ca
x2 +ba
x +
(b2a
)2=−ca
+
(b2a
)2
(x +
b2a
)2=
b2 − 4ac4a2
x +b2a
= ±√
b2 − 4ac2a
x =−b ±
√b2 − 4ac
2a
William McCallum Great Issues of Our Time: The Quadratic Formula
The quadratic formula
A number x satisfies
ax2 + bx + c = 0, a #= 0,
if, and only if,
x =−b +
√b2 − 4ac
2aor x =
−b −√
b2 − 4ac2a
.
Exercise
William McCallum Great Issues of Our Time: The Quadratic Formula
The quadratic formula
A number x satisfies
ax2 + bx + c = 0, a #= 0,
if, and only if,
x =−b +
√b2 − 4ac
2aor x =
−b −√
b2 − 4ac2a
.
ExerciseGive some simple conditions on the coefficients for a quadraticequation to have(a) two real roots Answer
(b) two positive roots. Answer
Examples Skip
William McCallum Great Issues of Our Time: The Quadratic Formula
The quadratic formula
A number x satisfies
ax2 + bx + c = 0, a #= 0,
if, and only if,
x =−b +
√b2 − 4ac
2aor x =
−b −√
b2 − 4ac2a
.
Answer(a) Make a and c have opposite signs. Answer to (b)
(b) Make a and c have the same sign, and make b negativebut large in magnitude.
Examples
William McCallum Great Issues of Our Time: The Quadratic Formula
The quadratic formula
A number x satisfies
ax2 + bx + c = 0, a #= 0,
if, and only if,
x =−b +
√b2 − 4ac
2aor x =
−b −√
b2 − 4ac2a
.
Answer(a) Make a and c have opposite signs.(b) Make a and c have the same sign, and make b negative
but large in magnitude.Examples
William McCallum Great Issues of Our Time: The Quadratic Formula
Another way of solving quadratic equations
For all numbers x ,
x2 + 10x − 39 = (x − 3)(x + 13) = 0.
Sox2 + 10x − 39 = 0
if, and only if,(x − 3)(x + 13) = 0.
The product of two numbers is zero if, and only if, one of themis zero, so either
x − 3 = 0 or x + 13 = 0.
That is, x = 3 or x = −13.
William McCallum Great Issues of Our Time: The Quadratic Formula
Another way of solving quadratic equations
For all numbers x ,
x2 + 10x − 39 = (x − 3)(x + 13) = 0.
Sox2 + 10x − 39 = 0
if, and only if,(x − 3)(x + 13) = 0.
The product of two numbers is zero if, and only if, one of themis zero, so either
x − 3 = 0 or x + 13 = 0.
That is, x = 3 or x = −13.
William McCallum Great Issues of Our Time: The Quadratic Formula
Another way of solving quadratic equations
For all numbers x ,
x2 + 10x − 39 = (x − 3)(x + 13) = 0.
Sox2 + 10x − 39 = 0
if, and only if,(x − 3)(x + 13) = 0.
The product of two numbers is zero if, and only if, one of themis zero, so either
x − 3 = 0 or x + 13 = 0.
That is, x = 3 or x = −13.
William McCallum Great Issues of Our Time: The Quadratic Formula
Viete’s formulae
Theorem (Viete’s formulae)The numbers r and s are the solutions to
x2 + bx + c = 0
if, and only if,r + s = −b and rs = c.
William McCallum Great Issues of Our Time: The Quadratic Formula
Viete’s formulae
Theorem (Viete’s formulae)The numbers r and s are the solutions to
x2 + bx + c = 0
if, and only if,r + s = −b and rs = c.
Proof of the “if” part.If r + s = −b and rs = c, then
x2 + bx + c = (x − r)(x − s).
Then the argument goes as in the previous slide. Exercise
William McCallum Great Issues of Our Time: The Quadratic Formula
Viete’s formulae
Theorem (Viete’s formulae)The numbers r and s are the solutions to
x2 + bx + c = 0
if, and only if,r + s = −b and rs = c.
ExerciseHow do you prove the “only if” part? That is, if r and s are thesolutions to x2 + bx + c, then r + s = −b and rs = c.
Answer Skip
William McCallum Great Issues of Our Time: The Quadratic Formula
Viete’s formulae
Theorem (Viete’s formulae)The numbers r and s are the solutions to
x2 + bx + c = 0
if, and only if,r + s = −b and rs = c.
Proof of “only if” part
Divide x − r into x2 + bx + c, so
x2 + bx + c = (x − r)q(x) + R.
Putting x = r we get R = 0. Then q(x) = x − t for some t , andthe only possibility is t = s.
William McCallum Great Issues of Our Time: The Quadratic Formula
Viete’s formulae and the quadratic formula
Ifx2 + bx + c = 0
then let
r =− b +
√b2 − 4c
2and s =
− b −√
b2 − 4c2
Exercise
William McCallum Great Issues of Our Time: The Quadratic Formula
Viete’s formulae and the quadratic formula
Ifx2 + bx + c = 0
then let
r =− b +
√b2 − 4c
2and s =
− b −√
b2 − 4c2
ExerciseGive an explanation, purely in terms of the structure of theexpressions, of why these two numbers satisfy
r + s = −b and rs = c.
Answer Skip
William McCallum Great Issues of Our Time: The Quadratic Formula
Viete’s formulae and the quadratic formula
Ifx2 + bx + c = 0
then let
r =− b +
√b2 − 4c
2and s =
− b −√
b2 − 4c2
AnswerWhen you add r and s, the plus and minus signs cancel.
When you multiply r and s, you get the difference of twosquares in the numerator,
(−b)2 − (√
b2 − ac)2 = b2 − (b2 − 4c) = 4c.
William McCallum Great Issues of Our Time: The Quadratic Formula
Viete’s formulae and the quadratic formula
Ifx2 + bx + c = 0
then let
r =− b
+√
b2 − 4c
2and s =
− b
−√
b2 − 4c
2
AnswerWhen you add r and s, the plus and minus signs cancel.
When you multiply r and s, you get the difference of twosquares in the numerator,
(−b)2 − (√
b2 − ac)2 = b2 − (b2 − 4c) = 4c.
William McCallum Great Issues of Our Time: The Quadratic Formula
Viete’s formulae and the quadratic formula
Ifx2 + bx + c = 0
then let
r =+
2and s =
−2
AnswerWhen you add r and s, the plus and minus signs cancel.When you multiply r and s, you get the difference of twosquares in the numerator,
(−b)2 − (√
b2 − ac)2 = b2 − (b2 − 4c) = 4c.
William McCallum Great Issues of Our Time: The Quadratic Formula
The quadratic formula in the 17th century
From the Oxford Museum of History of Science (Stephen Johnston, photo Bluebridge Farm Studio) Exercise
William McCallum Great Issues of Our Time: The Quadratic Formula
The quadratic formula in the 17th century
From the Oxford Museum of History of Science (Stephen Johnston, photo Bluebridge Farm Studio)
Exercise
What is going on here? Answer
William McCallum Great Issues of Our Time: The Quadratic Formula
What is going on here?
z + Cr = N :√qu :
Cq4
+ N : − C2
= r
Cr − z = N :
{C2 +√qu :C2 −
√qu :
}Cq4− N : = r
z − Cr = N :√qu :
Cq4
+ N : +C2
= r
William McCallum Great Issues of Our Time: The Quadratic Formula
What is going on here?
x2 + Cx = N :√qu :
Cq4
+ N : − C2
= x
Cx − x2 = N :
{C2 +√qu :C2 −
√qu :
}Cq4− N : = x
x2 − Cx = N :√qu :
Cq4
+ N : +C2
= x
William McCallum Great Issues of Our Time: The Quadratic Formula
What is going on here?
x2 + Cx = N :√qu :
C2
4+ N : − C
2= x
Cx − x2 = N :
{C2 +√qu :C2 −
√qu :
}C2
4− N : = x
x2 − Cx = N :√qu :
C2
4+ N : +
C2
= x
William McCallum Great Issues of Our Time: The Quadratic Formula
What is going on here?
x2 + Cx = N,√qu :
C2
4+ N : − C
2= x
Cx − x2 = N,
{C2 +√qu :C2 −
√qu :
}C2
4− N : = x
x2 − Cx = N,√qu :
C2
4+ N : +
C2
= x
William McCallum Great Issues of Our Time: The Quadratic Formula
What is going on here?
x2 + Cx = N,√qu :
C2
4+ N : − C
2= x
Cx − x2 = N,C2±√qu :
C2
4− N : = x
x2 − Cx = N,√qu :
C2
4+ N : +
C2
= x
William McCallum Great Issues of Our Time: The Quadratic Formula
What is going on here?
x2 + Cx = N,
√C2
4+ N − C
2= x
Cx − x2 = N,C2±
√C2
4− N = x
x2 − Cx = N,
√C2
4+ N +
C2
= x
William McCallum Great Issues of Our Time: The Quadratic Formula
ExampleIf
x2 + 10x − 39 = 0,
then
x =−10±
√256
2= −5±
√64 = 3,−13.
ExampleIf
x2 − 10x + 9 = 0
then
x =10±
√102 − 4× 9
2=
10±√
642
= 5± 4 = 1, 9.
Back to exercise
William McCallum Great Issues of Our Time: The Quadratic Formula
ExampleIf
x2 + 10x − 39 = 0,
then
x =−10±
√256
2= −5±
√64 = 3,−13.
ExampleIf
x2 − 10x + 9 = 0
then
x =10±
√102 − 4× 9
2=
10±√
642
= 5± 4 = 1, 9.
Back to exercise
William McCallum Great Issues of Our Time: The Quadratic Formula
ExampleIf
x2 + 10x − 39 = 0,
then
x =−10±
√256
2= −5±
√64 = 3,−13.
ExampleIf
x2 − 10x + 9 = 0
then
x =10±
√102 − 4× 9
2=
10±√
642
= 5± 4 = 1, 9.
Back to exercise
William McCallum Great Issues of Our Time: The Quadratic Formula