3. (a) Given that 2log(4−x)=log(x+8) show that
x2−9x+8=0
(b) (i) Write down the roots of the equation
x2−9x+8=0
(ii) State which of the roots in (b)(i) is not a solution of 2log(4−x)=log(x+8) giving a reason for your answer. - Edexcel - A-Level Maths: Pure - Question 6 - 2020 - Paper 2
Question 6
3. (a) Given that 2log(4−x)=log(x+8) show that
x2−9x+8=0
(b) (i) Write down the roots of the equation
x2−9x+8=0
(ii) State which of the roots in (b)(i) is not a ... show full transcript
Worked Solution & Example Answer:3. (a) Given that 2log(4−x)=log(x+8) show that
x2−9x+8=0
(b) (i) Write down the roots of the equation
x2−9x+8=0
(ii) State which of the roots in (b)(i) is not a solution of 2log(4−x)=log(x+8) giving a reason for your answer. - Edexcel - A-Level Maths: Pure - Question 6 - 2020 - Paper 2
Step 1
Given that 2log(4−x)=log(x+8) show that
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Answer
To prove that the equation x2−9x+8=0 is valid, start with the given equation:
Begin by rearranging the logarithmic equation:
2log(4−x)=log(x+8)
This can be rewritten using the properties of logarithms:
log((4−x)2)=log(x+8)
Since the logarithms are equal, the arguments must also be equal:
(4−x)2=x+8
Expand the left side:
(4−x)(4−x)=16−8x+x2
Set the equation to zero:
16−8x+x2=x+8x2−9x+8=0
This proves the equation as required.
Step 2
(b) (i) Write down the roots of the equation
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Answer
To find the roots of the quadratic equation x2−9x+8=0, we can factor it:
x2−9x+8=(x−1)(x−8)=0
From this, the roots are:
x=1
x=8
Step 3
(b) (ii) State which of the roots in (b)(i) is not a solution of 2log(4−x)=log(x+8) giving a reason for your answer.
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Answer
To determine which root is not a solution of 2log(4−x)=log(x+8), we substitute the roots x=1 and x=8 back into the logarithmic equation:
When x=1:
2log(4−1)=log(1+8)2log(3)=log(9)
This is true as log(9)=2log(3).
When x=8:
2log(4−8)=log(8+8)2log(−4)=log(16)
Here, log(−4) is undefined.
Thus, the root x=8 is not a solution because the logarithm of a negative number is not defined.
