A-Level Edexcel Maths Pure Circles: Given that $$2\log

Given that $$2\log_2{(x+15)} - \log_2{x} = 6$$ (a) Show that $$x^2 - 34x + 225 = 0$$ (b) Hence, or otherwise, solve the equation $$2\log_2{(x+15)} - \log_2{x} = 6$$ - Edexcel - A-Level Maths Pure - Question 7 - 2013 - Paper 6

Question 7

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Given that $$2\log_2{(x+15)} - \log_2{x} = 6$$ (a) Show that $$x^2 - 34x + 225 = 0$$ (b) Hence, or otherwise, solve the equation $$2\log_2{(x+15)} - \log_2{x} =... show full transcript

Worked Solution & Example Answer:Given that $$2\log_2{(x+15)} - \log_2{x} = 6$$ (a) Show that $$x^2 - 34x + 225 = 0$$ (b) Hence, or otherwise, solve the equation $$2\log_2{(x+15)} - \log_2{x} = 6$$ - Edexcel - A-Level Maths Pure - Question 7 - 2013 - Paper 6

Step 1

(a) Show that

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Answer

To show that $x^2 - 34x + 225 = 0$ , we start with the equation:

$2\log_2{(x+15)} - \log_2{x} = 6$ .

Rewrite the equation using logarithmic properties: $2\log_2{(x+15)} = \log_2{x} + 6$ . This can be expressed as: $\log_2{(x+15)^2} = \log_2{x} + \log_2{64}$ .
Then we can combine the logarithms: $\log_2{(x+15)^2} = \log_2{(64x)}$ .
Setting the arguments equal gives: $(x+15)^2 = 64x$ .
Expanding the left-hand side, we have: $x^2 + 30x + 225 = 64x$ .
Rearranging this equation leads to: $x^2 - 34x + 225 = 0$ , which is the required result.

Step 2

(b) Hence, or otherwise, solve the equation

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Answer

We now solve the quadratic equation:

$x^2 - 34x + 225 = 0$ .

Apply the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , where $a=1$ , $b=-34$ , and $c=225$ .
Calculate the discriminant: $b^2 - 4ac = (-34)^2 - 4(1)(225) = 1156 - 900 = 256$ .
Substitute back into the quadratic formula: $x = \frac{34 \pm \sqrt{256}}{2} = \frac{34 \pm 16}{2}$ .
This gives two potential solutions: $x = \frac{50}{2} = 25$ and $x = \frac{18}{2} = 9$ .

Thus, the solutions to the equation are $x = 25$ and $x = 9$ .

Given that $$2\log_2{(x+15)} - \log_2{x} = 6$$ (a) Show that $$x^2 - 34x + 225 = 0$$ (b) Hence, or otherwise, solve the equation $$2\log_2{(x+15)} - \log_2{x} = 6$$ - Edexcel - A-Level Maths Pure - Question 7 - 2013 - Paper 6

Worked Solution & Example Answer:Given that $$2\log_2{(x+15)} - \log_2{x} = 6$$ (a) Show that $$x^2 - 34x + 225 = 0$$ (b) Hence, or otherwise, solve the equation $$2\log_2{(x+15)} - \log_2{x} = 6$$ - Edexcel - A-Level Maths Pure - Question 7 - 2013 - Paper 6

(a) Show that

(b) Hence, or otherwise, solve the equation

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