Question

order the steps to solve the equation \\(\log(x^2 - 15) = \log(2x)\\) form 1 to 5. \\(x^2 - 2x - 15 = 0\\) potential solutions are -3 and 5 \\(x^2 - 15 = 2x\\) \\(x - 5 = 0\\) or \\(x + 3 = 0\\) \\((x - 5)(x + 3) = 0\\)

Explanation:

Step1: Drop log from both sides

$\log(x^2 - 15) = \log(2x) \implies x^2 - 15 = 2x$

Step2: Rearrange to standard quadratic form

$x^2 - 15 - 2x = 0 \implies x^2 - 2x - 15 = 0$

Step3: Factor the quadratic

$x^2 - 2x - 15 = (x - 5)(x + 3) = 0$

Step4: Set factors equal to 0

$(x - 5)(x + 3) = 0 \implies x - 5 = 0 \text{ or } x + 3 = 0$

Step5: Solve for potential solutions

$x - 5 = 0 \implies x=5; \ x + 3 = 0 \implies x=-3$

Answer:

1. $x^2 - 15 = 2x$
2. $x^2 - 2x - 15 = 0$
3. $(x - 5)(x + 3) = 0$
4. $x - 5 = 0$ or $x + 3 = 0$
5. Potential solutions are -3 and 5