Solve the Following Quadratic Equation by Factorization
Question:
\[ (x-5)(x-6)=\frac{25}{(24)^2} \]Solution
Given,
\[ (x-5)(x-6)=\frac{25}{576} \]Write the left side as a difference of squares:
\[ (x-5)(x-6) = \left(x-\frac{11}{2}\right)^2 -\left(\frac12\right)^2 \]Therefore,
\[ \left(x-\frac{11}{2}\right)^2-\frac14 = \frac{25}{576} \] \[ \left(x-\frac{11}{2}\right)^2 = \frac14+\frac{25}{576} \] \[ = \frac{144+25}{576} = \frac{169}{576} = \left(\frac{13}{24}\right)^2 \] \[ \left(x-\frac{11}{2}\right)^2 = \left(\frac{13}{24}\right)^2 \]Using \(a^2-b^2=(a-b)(a+b)\):
\[ \left(x-\frac{11}{2}-\frac{13}{24}\right) \left(x-\frac{11}{2}+\frac{13}{24}\right) =0 \] \[ \left(x-\frac{145}{24}\right) \left(x-\frac{119}{24}\right) =0 \]Therefore,
\[ x=\frac{145}{24} \] or \[ x=\frac{119}{24} \]