Solve $LaTeX: \displaystyle \log_{ 10 }(x + 15) + \log_{ 10 }(x + 105) = 3$

Using the product rule for logarithms gives $LaTeX: \displaystyle \log_{ 10 }(\left(x + 15\right) \left(x + 105\right))$ and rewriting in exponential form gives $LaTeX: \displaystyle \left(x + 15\right) \left(x + 105\right) = 1000$ expanding and setting the equation equal to zero gives $LaTeX: \displaystyle x^{2} + 120 x + 575 = 0$ . Factoring gives $LaTeX: \displaystyle \left(x + 5\right) \left(x + 115\right)=0$ . This gives two possible solutions $LaTeX: \displaystyle x=-115$ or $LaTeX: \displaystyle x=-5$ . $LaTeX: \displaystyle x=-115$ is an extraneous solution. The only soution is $LaTeX: \displaystyle x=-5$ .