A-Level Edexcel Maths Pure Simultaneous Equations: 3. (a) Given that 2log(4

3. (a) Given that 2log(4 - x) = log(x + 8) show that x^2 - 9x + 8 = 0 (b) (i) Write down the roots of the equation x^2 - 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 5 - 2020 - Paper 2

Question 5

3.-(a)-Given-that-2log(4---x)-=-log(x-+-8)-show-that-x^2---9x-+-8-=-0--(b)-(i)-Write-down-the-roots-of-the-equation-x^2---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 5-2020-Paper 2.png

3. (a) Given that 2log(4 - x) = log(x + 8) show that x^2 - 9x + 8 = 0 (b) (i) Write down the roots of the equation x^2 - 9x + 8 = 0 (ii) State which of the roots in... show full transcript

Worked Solution & Example Answer:3. (a) Given that 2log(4 - x) = log(x + 8) show that x^2 - 9x + 8 = 0 (b) (i) Write down the roots of the equation x^2 - 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 5 - 2020 - Paper 2

Step 1

Given that 2log(4 - x) = log(x + 8) show that x^2 - 9x + 8 = 0

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Answer

To demonstrate that the relationship holds, we start with the given equation:

2log(4 - x) = log(x + 8)

Using the property of logarithms that states $a \cdot log(b) = log(b^a)$ , we can rewrite the left side as:

log((4 - x)^2) = log(x + 8)

Since the logarithm is a one-to-one function, we can set the arguments equal to each other:

(4 - x)^2 = x + 8

Expanding the left side gives:

16 - 8x + x^2 = x + 8

Now, rearranging the equation leads us to:

x^2 - 9x + 16 - 8 = 0$$ This simplifies to:

x^2 - 9x + 8 = 0$$

Thus, we have shown that the equation holds.

Step 2

(b) (i) Write down the roots of the equation x^2 - 9x + 8 = 0

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Answer

To find the roots of the quadratic equation $x^2 - 9x + 8 = 0$ , we can factor the equation:

(x - 1)(x - 8) = 0

This gives us the roots:

$x = 1$
$x = 8$

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, we substitute the roots back into the original logarithmic equation:

For $x = 1$ :
$2log(4 - 1) = log(1 + 8)$
Simplifies to:
$2log(3) = log(9)$
This holds true (as $2log(3) = log(3^2) = log(9)$ ).
For $x = 8$ :
$2log(4 - 8) = log(8 + 8)$
This simplifies to:
$2log(-4) = log(16)$
Since logarithms of negative numbers are undefined, $x = 8$ is not a valid solution.

Thus, $x = 8$ is the root that does not satisfy the given logarithmic equation.

Given that 2log(4 - x) = log(x + 8) show that x^2 - 9x + 8 = 0

(b) (i) Write down the roots of the equation x^2 - 9x + 8 = 0

(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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