AQA GCSE Maths

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GCSEMathsAQATopic Questions2. AlgebraSolving Quadratic Equations

Solving Quadratic Equations

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1
Sme Calculator1 mark

Faisal tries to solve left parenthesis x space plus space 2 right parenthesis left parenthesis x space – space 7 right parenthesis space equals space 0

Here is his working.

table row cell space space space space space space space space space space space space space space space space space space space left parenthesis x space plus space 2 right parenthesis space end cell equals cell space 0 space space space space or space space space space left parenthesis x space – space 7 right parenthesis space equals space 0 space end cell row cell Answer space space space space space space space space space space space space space space space space space x space end cell equals cell space 2 space space space space or space space space space space x space equals space 7 end cell end table

Give a reason why his answer is wrong.

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2
Sme Calculator2 marks

Work out the two roots of  left parenthesis 7 x space plus space 1 right parenthesis left parenthesis 2 x space – space 3 right parenthesis space equals space 0

Circle both roots.

negative 1 over 7 1 over 7 negative 3 over 2 3 over 2

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3
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Circle the two roots of   open parentheses x minus 5 close parentheses open parentheses x plus 3 close parentheses equals 0

-5 -3 3 5

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4
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Jo wants to work out the solutions of      x squared space plus space 3 x space – space 5

She says,

   ‘‘The solutions cannot be worked out because 
            x squared space plus space 3 x space – space 5 does not factorise to left parenthesis x space plus space a right parenthesis left parenthesis x space plus space b right parenthesis where a and b are integers.’’

Is Jo correct?
Tick a box.

begin mathsize 26px style square end style   Yes                  begin mathsize 26px style square end styleNo

Give a reason for your answer.

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5
Sme Calculator1 mark

Circle the two roots of left parenthesis 2 x space plus space 3 right parenthesis left parenthesis 5 x space – space 2 right parenthesis space equals space 0

negative 3 over 2 negative 2 over 5 2 over 5 3 over 2

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6
Sme Calculator1 mark

Circle the equation with roots 4  and negative 8.

4 x left parenthesis x space – space 8 right parenthesis space equals space 0

left parenthesis x space – space 4 right parenthesis left parenthesis x space plus space 8 right parenthesis space equals space 0
x squared space – space 32 space equals space 0 left parenthesis x space plus space 4 right parenthesis left parenthesis x space – space 8 right parenthesis space equals space 0

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1
Sme Calculator3 marks

Solve     x squared space minus space 6 x space minus space 8 space equals space 0

Write your answer in the form a space plus-or-minus space square root of b where a spaceand b spaceare integers.

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2
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Alison is using the quadratic formula to solve a quadratic equation.
She substitutes values into the formula and correctly gets.

x space equals space fraction numerator negative 7 space plus-or-minus space square root of 49 space minus 32 end root over denominator 4 end fraction

Work out the quadratic equation that Alison is solving.
Give your answer in the form a x squared plus space b x space plus space c space equals space 0, where a comma space b and c are integers.

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3
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Solve   3 x squared space minus space 4 x space minus 2 space equals space 0

Give your solutions correct to 3 significant figures.

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4
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Solve   3 x squared space minus space 5 x space minus 1 space equals space 0

Give your solutions correct to 3 significant figures.

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5
Sme Calculator3 marks

Solve   2 x squared space plus space 3 x space minus 7 space equals space 0

Give your solutions correct to 2 decimal places.

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6
Sme Calculator3 marks

q6-easy-2-10-solving-quadratic-equation-edexcel-gcse-maths

The area of square A B C D is 10 cm2.
Show that space x squared space plus space 6 x space equals space 1

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7
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Solve 5 x to the power of 2 space end exponent equals space 10 x space plus space 4

Give your answers to 2 decimal places.

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8a
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2 x squared space minus space 6 x space plus space 5 can be written in the form a left parenthesis x space minus space b right parenthesis squared plus space c

where a comma space b andspace c space are positive numbers.

Work out the values of  a comma space b and c.

table attributes columnalign right center left columnspacing 0px end attributes row a equals cell................... end cell row b equals cell................... end cell row c equals cell................... end cell end table

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8b
Sme Calculator3 marks

Using your answer to part (a), or otherwise, solve  2 x squared space – space 6 x space plus space 5 space equals space 8.5]

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9a
Sme Calculator3 marks

Here are two pieces of work.

For each one, describe the error made and give the complete correct solution.

Question:

Solve by factorisation.

3 x squared space – space 2 x space – space 5 space equals space 0

Solution:

left parenthesis 3 x space plus space 5 right parenthesis left parenthesis x space – space 1 right parenthesis space equals space 0

Therefore x space equals space minus 5 divided by 3 space or space x space equals space 1

Error: ........................................................

Correct solution:

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9b
Sme Calculator3 marks

Question:
Solve, giving your answers correct to 3 significant figures.

2 x squared space – space 8 x space plus space 3 space equals space 0

Solution:

x equals negative open parentheses blank to the power of minus 8 close parentheses plus-or-minus fraction numerator square root of open parentheses blank to the power of minus 8 close parentheses squared minus 4 cross times 2 cross times 3 end root over denominator 2 cross times 2 end fraction

Therefore x space equals space 6.42 or x space equals space 9.58

Error: .....................................................................

Correct solution:

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10
Sme Calculator3 marks

Solve by factorisation.

5 x squared plus 7 x plus 2 equals 0

x = ............................ or x = ............................

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11
Sme Calculator3 marks

Solve by factorisation.

2x2 − 19x − 33 = 0

x = ................... or x = .................. 

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12
Sme Calculator7 marks
i)
Write x squared plus space 4 x minus space 16 in the form open parentheses x plus a close parentheses squared space minus space b.

[3]

ii)
Solve the equation x squared plus space 4 x minus space 16 space equals 0.
Give your answers in surd form as simply as possible.

x = ........... or begin mathsize 16px style x end style = .......... [4]

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1
Sme Calculator2 marks

Solve    open parentheses x minus 2 close parentheses squared space equals 3

Give your solutions correct to 3 significant figures.

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2
Sme Calculator3 marks

Solve the equation 3 x squared space plus space 4 x long dash space 12 space equals space 0

Give your solutions correct to 2 decimal places.

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3
Sme Calculator3 marks

Solve 3 x squared space plus space 6 x space minus 2 space equals 0

Give your solutions correct to 2 decimal places.

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4
Sme Calculator3 marks

Solve    3 x squared space long dash x long dash space 1 space equals space 0

Give your solutions correct to 2 decimal places.

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5
Sme Calculator3 marks

Solve space x squared space long dash space 5 x plus space 3 space equals space 0

Give your solutions correct to 3 significant figures.

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6
Sme Calculator1 mark

Liz is asked to solve the equation 3 x squared space plus space 8 space equals space 83

Here is her working.

table row cell 3 x squared space plus space 8 space end cell equals cell space 83 end cell row cell 3 x squared space end cell equals cell space 75 end cell row cell x to the power of 2 space end exponent end cell equals cell space 25 end cell row cell x space end cell equals cell space 5 end cell end table

Explain what is wrong with Liz's answer.

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7
Sme Calculator2 marks

The only solution to x squared space plus space b x space plus space c space equals space 0    is    x space equals space 5

Work out the values of b and c.

b equals.................... c equals....................

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8a
Sme Calculator1 mark

Write x left parenthesis 3 x space – space 9 right parenthesis space equals space 4 in the form  a x squared plus b x plus c equals 0 where a comma space b and space c spaceare integers.

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8b
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Solvespace x left parenthesis 3 x space – space 9 right parenthesis space equals space 4

Give your answers to 2 decimal places.

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9
Sme Calculator3 marks

Solve x squared space minus space x space minus 12 space equals space 0

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10
Sme Calculator3 marks

Without expanding any brackets,

   show how to work out the exact solutions of    9 left parenthesis x space plus space 3 right parenthesis squared space equals space 4

Give the solutions.

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11
Sme Calculator3 marks

Solve by factorising.

x squared plus 9 x plus 20 equals 0

x= ....................... or x = .......................

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12
Sme Calculator4 marks

Solve this equation algebraically.
Give your solutions correct to 2 decimal places.

3x2 + 8x - 5 = 0

x = .................... or x = ....................

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13
Sme Calculator4 marks

Solve.

x squared space minus space 6 x space plus space 15 space equals space 3 x space minus space 5

x space equals space.............. or x space equals space..............

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14
Sme Calculator2 marks

Solve.

   3 x squared space equals space 75

x = .................

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15
Sme Calculator3 marks

Solve  x squared space – space 21 x space plus space 20 space equals space 0

Show your working clearly.

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16a
Sme Calculator2 marks

Factorise  x squared space plus space 2 x space – space 24

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16b
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Hence solve  x squared space plus space 2 x space – space 24 space equals space 0

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1
Sme Calculator3 marks

Solve    x squared space equals space 4 open parentheses x minus 3 close parentheses squared

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2a
Sme Calculator3 marks

Solve         2 x squared space plus space 9 x space minus space 7 space equals space 0

Give your solutions correct to 3 significant figures.

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2b
Sme Calculator2 marks

Solve      2 over y squared space plus space 9 over y space minus space 7 space equals space 0

Give your solutions correct to 3 significant figures.

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3
Sme Calculator5 marks

Given that

2 x minus 1 space colon thin space x minus 4 space equals space 16 x space plus 1 space colon space 2 x space minus 1

find the possible values of x.

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4a
Sme Calculator2 marks

The diagram shows a trapezium.

q4-hard-2-10-solving-quadratic-equation-edexcel-gcse-maths

All the measurements are in centimetres.

The area of the trapezium is 351 cm2.

Show that  2 x squared space plus space x space minus space 351 space equals space 0

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4b
Sme Calculator3 marks

Work out the value of x.

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5
Sme Calculator4 marks

Given that

x squared space colon space open parentheses 3 x space plus space 5 close parentheses space equals space 1 space colon space 2   

find the possible values of  x.

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6
Sme Calculator5 marks

q5-hard-2-10-solving-quadratic-equation-edexcel-gcse-maths

The lengths of the sides are in centimetres.

The area of triangle T1 is equal to the area of triangle T2.

Work out the value of x, giving your answer in the form  a space plus space square root of b  where a and b are integers.

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7
Sme Calculator5 marks

Solve  fraction numerator 5 over denominator 4 x plus 1 end fraction equals fraction numerator 2 x over denominator x squared plus 3 end fraction

Give your solutions to 3 significant figures.
You must show your working.

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8
Sme Calculator6 marks

y space equals space 6 x to the power of 4 space plus space 7 x squared and x space equals space square root of w space plus 1 end root

Find the value of w when space y space equals space 10.
Show your working.

w = ................ 

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9
Sme Calculator5 marks

Given that M space equals fraction numerator 18 to the power of 4 n end exponent cross times 2 to the power of 3 open parentheses n squared minus 6 n close parentheses end exponent cross times 3 to the power of 2 open parentheses 1 minus 4 n close parentheses end exponent over denominator 12 squared end fraction 

find the values of n for which M space equals space 2

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10a
Sme Calculator3 marks

The diagram shows a trapezium.

q15-paper2h-jan2019-edexcel-igcse-maths

All measurements shown on the diagram are in centimetres.
The area of the trapezium is 133 space cm squared

Show that 8 x squared space – space 6 x space – space 275 space equals space 0

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10b
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Find the value of x.

Show your working clearly.

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11a
Sme Calculator2 marks

Express 7 space – space 4 x space – space x squared in the form p space – space left parenthesis x space plus space q right parenthesis squared where p and q are constants.

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11b
Sme Calculator3 marks

Use your answer to part (a) to solve the equation 7 space – space 4 left parenthesis space y space plus space 3 right parenthesis space – space left parenthesis space y space plus space 3 right parenthesis squared equals space 0

Give your solutions in the form e space plus-or-minus square root of f where e andspace f are integers.

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12
Sme Calculator3 marks

6 x squared equals 7 x y plus 20 y squared  where  x greater than 0 and  y greater than 0

Find the ratio  x colon y
 

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