Solving Quadratic Equations|Maths:149 | Class:12th

Practice Questions and Solutions

Non-Calculator

Q1.
(a) Factorise x2 + 5x − 24

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Answer _________________________________________
(2)

(b) Solve x2 + 5x − 24 = 0

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Answer _________________________________________
(1)
(Total 3 marks)

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 Q2.
I am thinking of two numbers.

The first number is x.

The second number is 7.5 more than x.

(a) Write down an expression, in terms of x, for the second number.

Answer _________________________________________
(1)

(b) For the two numbers,

the product is double the sum.

Work out the numbers I could be thinking of.

Give both possible pairs of answers.

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Answer _______________ and _______________

or

_______________ and _______________

(5)
(Total 6 marks)

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 Q3.
The graph of y = x2 + 2x − 3 is drawn below.

Draw an appropriate straight line on the graph to work out the approximate solutions of

x2 + x − 3 = 0
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Answer _________________________________________

y = x2 + 2x − 3

(Total 3 marks)

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 Calculator

Q4.

The expression simplifies to

Work out the value of b.

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b = ______________________________________
(Total 3 marks)

Q5.
The diagram shows a rectangle.

The area of the rectangle is 90 cm2.

Set up and solve a quadratic equation to work out the value of x.

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x = ___________________________________ cm
(Total 5 marks)

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 Q6.
(a) Expand and simplify (x + 5)(x − 4)

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Answer _________________________________________
(2)

(b) Solve (x − 8)(x + 7) = 0

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Answer _________________________________________
(1)
(Total 3 marks)

Q7.

(a) Factorise x2 − 9x + 20
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Answer _________________________________________
(2)

(b) Solve x2 − 9x + 20 = 0
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Answer _________________________________________
(1)
(Total 3 marks)

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 Q8.
The area of this rectangle is 28 cm2

Not drawn accurately

Work out the perimeter of the rectangle.

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Answer ______________________________________ cm
(Total 7 marks)

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 Q9.
(a) Simplify fully 5x2 × 3y4 × 2x × y3

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Answer _________________________________________
(2)

(b) Expand and simplify (x + 7)(x – 3)

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Answer _________________________________________
(2)

(c) Solve (x – 8)(x + 2) = 0

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Answer _________________________________________
(1)

(d) Factorise 8x2y + 6xy2

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Answer _________________________________________
(2)
(Total 7 marks)

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 Solutions | Solving Quadratic Equations|Maths:149
Mark schemes

Q1.
(a) (x + a)(x + b)
where ab = ± 24
M1

(x + 8)(x – 3)
either order
A1

(b) (x =) – 8 and (x =) 3
ft their part (a)
B1 ft
[3]

Q2.
(a) x + 7.5 or 7.5 + x

x+7
B1

(b) x(x + 7.5) = 2(x + x + 7.5)

ft their x + 7.5 from (a) in the form x + c for all 4 method
marks
M1

x2 + 7.5x = 4x + 15
M1

x2 + 3.5x − 15 = 0

or

2x2 + 7x − 30 = 0
M1

(2x − 5)(x + 6) (= 0)
M1

2.5 and 10
either order but in correct pairs

and

−6 and 1.5
SC1 one correct pair
A1
[6]

Q3.

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 (x2 + 2x − 3) − (x2 + x − 3)
Or attempt to ‘balance’ equations
M1

y=x
A1

− 2.3 and 1.3

ft if M awarded and their line drawn
A1ft
[3]

Q4.
(x − 3)(x + 3)
Substitutes any value for x into both
expressions but not x = 0
M1

(x − 3)(x + 5)
Sets up a correct equation in b
M1dep

(b =) 2 or x 2 + 2x − 15
A1
[3]

Q5.
(x + 4)(x − 5) (= 90)
M1

x2 + 4x − 5x − 20 (= 90)
Allow 1 error
M1

x2 − x − 110 (= 0)
Collecting their 4 terms and 90
dependent on 2nd M1 only
M1dep

(x + 10)(x − 11)
(x + a)(x + b) where ab = ± their 110
Use of formula – allow one error
M1

11
Note: 11 and − 10 implies M4A0
A1
[5]

Q6.
(a) x2− 4x + 5x − 20

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 Allow one error
M1

x2 + x − 20
A1

(b) 8 and −7
B1
[3]

Q7.
(a) (x − 4)(x − 5)
B1 for (x − a)(x − b) where ab = 20
or a + b = −9
B2

(b) 4 and 5
ft their part (a) provided two brackets
B1ft
[3]

Q8.
(x + 2)(6x – 1) = 28
M1

6x2 – x + 12x – 2 = 28
Allow one error
M1dep

6x2 +11x – 30 (= 0)
Collect terms to one side, ft their four terms
M1dep

(3x + 10)(2x – 3) (= 0)
A1

(x = – and) x = 1.5
oe
ft their two brackets
B1ft

2(6 × 1.5 – 1 + 1.5 + 2)

or 14 × 1.5 + 2
2(6x – 1 + x + 2)
or 14x + 2
M1

23

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 (and x = – discarded)
May be implied
B1ft
[7]

Q9.
(a) 30x3y7
B1 for two correct terms
B2

Additional Guidance

Do not ignore fw for B2

30 × x3 × y7
B1

30 × x3y7
B1

x3y730
B1

7x3 × 4y7
B1

Do not allow addition sign,

eg 10x3 + 3y7
B0

(b) x2 – 3x + 7x – 21
Allow one error
M1

x2 + 4x – 21
A1

Additional Guidance

Do not ignore fw unless attempting to solve the equation

x2 – 3x – 21 or x2 + 7x – 21 (one error)
M1A0

x2 – 21 (two errors)
M0A0

x2 – 4x – 21 with no other working (two errors)

M0A0

(c) 8 and –2

or x = 8 and x = –2

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 Any order
B1

(d) 2xy (4x + 3y)

B1 for a correct partial factorisation
x (8xy + 6y2)
y (8x2 + 6xy)
2 (4x2y + 3xy2)
2x (4xy + 3y2)
2y (4x2 + 3xy)
xy (8x + 6y)
B2
[7]

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