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20
Quadratic equations
Links to:
Higher Student Book
Ch20, pp.xx–xx
Key Points
B A
Difference of two squares
An expression of the form x  b , where x and b are
numbers or algebraic terms, is called the difference of
two squares. In general, x2  b2  b2 (x  b)(x  b).
2
2
B A
Factorising quadratic expressions
To factorise an expression of the form x2  bx  c you
must find two numbers whose product is c and sum is b.
To factorise an expression of the form ax2  bx  c you
must first consider the factors of the coefficient of x2 and
then the other terms.
Solving quadratic equations
B
1
The solution to the quadratic equation ax2  bx  c  0,
when a  0 is
________
b  √b2  4ac
_______________
x
2a
• Completing the square
First write the quadratic in the form (x  b)2  c, then
rearrange the equation to find the exact value of x.
A*
The discriminant
The discriminant is the expression b  4ac under the
square root in the quadratic formula.
2
You can solve quadratic equations by
• If b2  4ac  0, there are two distinct solutions to the
quadratic equation.
• Rearranging
Rearrange the equation to make the unknown the subject.
• If b2  4ac  0, there are no real solutions to the
quadratic equation.
• Factorising
Rearrange the equation so all the terms are on one side,
then factorise it. This way one of the pair of brackets must
be equal to zero.
• If b2  4ac  0, there is one solution
(sometimes called a repeated root).
20.1
B
B A A*
• Using the quadratic formula
Factorising the difference of two squares
a Copy and complete this statement.
x2  225  (x 
)(x 
)
b Check your answer to part a by expanding the brackets.
a a2  81
b b2  169
2
Factorise the following.
3
I think of a number, I square it and subtract 1.
c x2  900
a Write down an algebraic expression to illustrate this.
AO 2
b Factorise your answer to part a.
A
4
A
5
Factorise the following.
a 25a2  49
b 16b2  25
c 400x2  9
d 121x2  81
Factorise each of the following by first taking out a common factor.
a 18a2  98
b 75b2  27
c 5x2  500
d 400x2  4
AO 2
20.2
B
B
1
2
Factorising quadratics of the form x2  bx  c
Factorise the following.
a x2  3x  2
b x2  7x  10
c x2  11x  30
d x2  7x  10
e x2  20x  100
f x2  11x  10
g x2  2x  8
h x2  5x  14
i x2  5x  6
j x2  x  6
k x2  2x  24
l x2  10x  200
Find two factors for the expression e2  3e  10.
AO 2
68
20 Quadratic equations
20.3
1
Solving quadratic equations
Solve these equations by rearranging.
a x2  400
2
b x2  7  42
Solve these equations by rearranging.
a 3x2  12  63
3
b 2x2  7x2  80
c 122  4t2  86
b (x  2)2  12  37
(x  3)2
c _______  1
4
Find the roots of
a 3(x  1)2  48
4
B
x2  16
d ____
2.25
c 3x2  27
A laser etches a rectangle onto the side of a diamond.
B
The length of the rectangle is four times its width.
The area of the rectangle is 0.0064 cm2.
a Using x for the width of the rectangle, form and solve an equation.
AO 2
b What is the length of the rectangle in mm?
a x2  4x  0
b b2  5b  0
c 5x2  15x  0
d 2x2  12x  0
e 3b2  b  0
f 4x  8x2
L F
U
She remembers that the length of the field is three times the width.
She also knows that the field has an area of 97 200 m2.
Unfortunately she can’t remember the dimensions of the field.
NA
NCTIO
NCTIO
Last year a farmer put a fence around one of her fields.
U
6
B
Solve the following quadratic equations by factorising.
NA
5
B
L F
Form and solve a quadratic equation to work out the dimensions of the field.
7
The product of two consecutive even numbers is 288.
AO 3
Form and solve a quadratic equation to find the two numbers.
20.4
1
Factorising quadratics of the form ax2  bx  c
Factorise each quadratic expression.
a 3x2  11x  6
2
5
b 8x2  34x  21
c 6x2  x  15
Factorise each quadratic expression. Start by taking out a common factor.
a 2x2  14x  24
4
A
c 7x2  13x  2
Factorise each quadratic expression.
a 4x2  16x  15
3
b 5x2  46x  9
b 2x2  20x  42
c 15x2  50x  40
Solve the following for x. Leave answers as fractions where necessary.
a 2x2  3x  1  0
b 3x2  5x  2  0
c 4x2  9x  9  0
d 5x2 2  11x
e x  15  6x2
f 7x2  2  13x
3x � 2
a Write down an algebraic expression for the area of the rectangle.
2
The area of the rectangle is 86 cm .
2x � 3
A
AO 2
b Work out the value of x.
20 Quadratic equations
69
A
6
Three times my square is equal to my value added to 80.
AO 3
A*
I am a negative number.
What number am I?
7
Next year Ian will be nine times the age of his son Joe.
Let x represent Joe’s age next year.
a Write down an algebraic expression for
i Ian’s age next year
ii Joe’s age this year
iii Ian’s age this year.
The product of their ages this year is 176.
b Form and solve an equation to work out Joe’s age next year.
AO 2
c How old is Ian this year?
20.5
A
A*
1
2
Use the quadratic formula to solve the following. Leave your answers in surd form.
a x2  6x  1  0
b 2x2  6x  3  0
c 3x2  8x  6  0
d 5x2  5x  1
e 7  11x  6x2
f 7x2  4x  29
For each of these quadratic equations, decide if there are zero, one or two solutions.
a 2x2  15x  18  0
b x2  x  1  0
c 3x2  7x  8  0
d 3x2  6x  3  0
16
g x  8 ___
x
e 7x  5x2  3
3  4x
h _____
x2
f (2x  4)2  6x  10
10
i _____________
1
(x  2)(x  3)
20.6
A*
1
Using the quadratic formula
Completing the square
Write each expression in completed square form.
a x2  6x  1
2
c x2  12x  3
Write each algebraic expression in the form (x  p)2  q, giving the values of p and q.
a x2  4x  1
3
b x2  4x  3
b x2  8x  9
c x2  6x  4
Solve the following quadratic equations by completing the square.
Give your answers in both surd form and to two decimal places.
70
a x2  8x  2  0
b 2x2  3x  4  0
d 3x2  1  6x
e (2x  2)2  2x2  11
20 Quadratic equations
c 3x2  6x  1  0
7  2x
1
f ______________
(4x  1)(x  1)