Given that $$2 \, ext{log}_2 (x + 15) - ext{log}_2 x = 6$$ (a) Show that $$x^2 - 34x + 225 = 0$$ (b) Hence, or otherwise, solve the equation $$2 \, ext{log}_2 (x + 15) - ext{log}_2 x = 6$$ - Edexcel - A-Level Maths Pure - Question 8 - 2013 - Paper 6

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

$2 \, ext{log}_2 (x + 15) - ext{log}_2 x = 6$.

Step 1: Rewrite the left side:

Using the property of logarithms that states $a \, ext{log}_b c = ext{log}_b (c^a)$, we can rewrite the equation as:

$ext{log}_2 ((x + 15)^2) - ext{log}_2 x = 6$.

Step 2: Apply the logarithmic subtraction rule:

We can express this as:

$ext{log}_2 \left( \frac{(x + 15)^2}{x} \right) = 6$.

Step 3: Exponentiate both sides:

This gives us:

$\frac{(x + 15)^2}{x} = 2^6$, which simplifies to:

$\frac{(x + 15)^2}{x} = 64$.

Step 4: Clear the fraction:

Multiplying both sides by $x$ results in:

$(x + 15)^2 = 64x$.

Step 5: Expand and rearrange:

Expanding the left side, we have:

$x^2 + 30x + 225 = 64x$.

Now, rearranging gives:

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

Thus, we have shown what was required.