8. Solving quadratic equations

Name ……………………………………………………
Class …………
Date ………………….
8. Solving quadratic equations.
1
Multiply out and simplify:
a (c + 3)(c + 5) =
b (t + 6)(t + 2) =
c (x – 7)(x + 4) =
d (m + 1)(m – 8) =
e (f – 5)(f – 5) =
f (b – 2)(b + 4) =
g (y – 9)(y + 6) =
h (n + 7)(n – 1) =
i (a + 3)2 =
j (p – 4)2 =
k (2c + 6)(c + 5) =
l (3q + 1)(q – 5) =
m (4d – 3)(d – 2 ) =
n (5j + 2)(3j – 5) =
o (2x + 5)2 =
p (3y – 4)2 =
q (5w – 3)(4w + 2) =
r (2y – 3)(2y + 1) =
2
a Multiply out and simplify (4c – 1)(2c + 3)
b Show how you could use your answer to part a) to work out 39  23
3
a Write an expression for the area of this rectangle, using brackets.
b Expand the brackets and simplify.
1
Name ……………………………………………………
4
Class …………
Date ………………….
Factorise each of these expressions:
a x2 + 5x + 6 =
b d2 + 11d + 24 =
c p2 + 8p + 15 =
d g2 – 7g + 12 =
e a2 – 7a + 10 =
f n2 + 4n – 12 =
g m2 + 7m – 8 =
h k2 – k – 6 =
i b2 – 5b – 24 =
j h2 – 12h + 35 =
k v2 + 4v + 4 =
l t2 – 10t + 25 =
m h2 + 13h + 36 =
n q2 – 8q + 12 =
o x2 – 36 =
p c2 – 81 =
q z2 – 5z + 4 =
r t2 – 2t – 15 =
5
Factorise these:
a 2x2 + 5x + 2 =
b 3c2 + 14c + 8 =
c 2m2 + 13m + 15 =
d 15a2 + 18a + 3 =
e 2n2 – 5n + 3 =
f 6p2 – 11p + 4 =
g 6b2 – 7b – 20 =
h 6t2 + 19t + 10 =
i 3c2 + 19c – 14 =
j 25t2 – 49 =
k 6x2 – 13x + 6 =
l 9y2 – 3y – 20 =
2
Name ……………………………………………………
Class …………
Date ………………….
m 9v2 – 1 =
n 6k2 – 10k – 4 =
o 10w2 + w – 3 =
p 8d2 – 22d + 12 =
q 10m2 – 4m – 6 =
r 3x2 – 27 =
s 18a2 – 8 =
t 20p2 – 5 =
6
a
Simplify these algebraic fractions:
pq
5p
=
b 8mn =
12n
c 15x =
3y
d 12a2 =
8a
e 4(w  3) =
8w
f
3ab
=
a(b  1)
g abc2 =
ab
h 2m  4 =
m2
i
p 2  6p
p6
j
y2  9 =
y  5y  6
=
2
k
l
q 2  5q  14
q 2  6q  8
2c 2  9c  5
6c 2  c  2
=
=
3
Name ……………………………………………………
m
b 2  7b  10
b 2  25
=
n
2k 2  2k  4
4k 2  2k  12
=
o
3n 2  7n  2
2n 2  3n  2
p
8 x 2  20 x  12
6 x 2  15 x  9
7
Class …………
Date ………………….
=
=
Solve these quadratic equations:
a (t + 6)(t – 4) = 0
b (p – 3)(2p + 7) = 0
c (d – 2)2 = 0
d (q + 5)(3q – 1) = 0
e b2 + 3b – 10 = 0
f a2 + 7a + 10 = 0
g z2 – z – 20 = 0
h x2 – 2x + 1 = 0
i y2 – 5y + 6 = 0
j c2 + 9c + 14 = 0
k k2 + 4k = –3
l h2 – 11h = –30
m v2 – 11v = 26
n x2 + x = 6
o r2 – 36 = 0
p g2 = 16
q 2w2 – 18w = 0
r t2 + 10t = 0
s 3x2 – 3x – 4 = 0
t 6k2 – 19k + 10 = 0
u 8m2 – 13m – 6 = 0
v 12b2 + 26b + 12 = 0
w 8c2 – 49c + 6 = 0
x 10n2 – 11n – 6 = 0
y 20 q2 – 7q = 6
z 8g2 + 6g = 35
8 A triangle has a height of (x + 1) cm and a base of 2x cm.
Write down an expression for the area of the triangle and
simplify it.
The area of the triangle is 20 cm2.
Find the value of x.
9 The quadratic equation y2 + 7y – n = 0 has a solution y = –9.
a Work out the value of n.
b Find the other solution for y.
4
Name ……………………………………………………
Class …………
Date ………………….
10 Sanjay thinks of a number, squares it and then subtracts twice the original number.
The result is 63.
Write down a quadratic equation and solve it to find the two possible numbers that
Sanjay could have started with.
11 Eleanor has to solve the equation n2 + 5n – 6 = 18
Here is her solution:
(n – 1)(n + 6) = 18
either n – 1 = 0 when n = 1
or n + 6 = 0 when n = –6
Eleanor checks her first answer.
12 + 5 × 1 – 6 = 1 + 5 – 6 = 0
The answer does not equal 18, so she knows she has made a mistake.
a What is Eleanor’s mistake?
b Work out the correct solutions to the equation.
12 The rectangle ABCD has a perimeter of 14 cm and side BD = x cm.
a Write down, in terms of x, an expression for the area of the rectangle.
b The area of the rectangle is 12 cm2.
Form an equation in x and solve it to find the length and the width of the rectangle.
13 y 2 + 6x + 4 = (y + 3)2 – a
Work out the value of a.
5
Name ……………………………………………………
Class …………
Date ………………….
14 p 2 – 10p + 40 = (p – 5)2 + b
Work out the value of b.
15 Solve these equations by first completing the square.
Give your answers in surd form.
a m2 + 6m + 4 = 0
b a2 + 4a – 3 = 0
c n2 + 8n + 13 = 0
d g2 + 10g – 3 = 0
e x2 – 4x + 2 = 0
f y2 – 8y – 1 = 0
g p2 + 2p – 5 = 0
h v2 – 6v + 1 = 0
i 5q2 + 20q – 10 = 0
j 3b2 – 6b – 12 = 0
k r2 + 6r + 1 = 0
l t2 – 4t – 11 = 0
16 You are given the identity x2 + px + 16 = (x + q)2
Work out the values of p and q.
17 You are given the identity y2 – ay + 100 = (y – b)2
Work out the values of a and b.
6

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