Determine Whether 16x² = 24x + 1 Has Real Roots and Find the Roots
Question
Determine whether the given quadratic equation has real roots and if so, find the roots:
\[ 16x^2=24x+1 \]Solution
Write the equation in standard form:
\[ 16x^2-24x-1=0 \]
\[ a=16,\quad b=-24,\quad c=-1 \]
Find the discriminant:
\[ D=b^2-4ac \]
\[ D=(-24)^2-4(16)(-1) \]
\[ D=576+64 \]
\[ D=640 \]
Since \(D>0\), the equation has two distinct real roots.
\[ x=\frac{-b\pm\sqrt{D}}{2a} \]
\[ x=\frac{24\pm\sqrt{640}}{32} \]
\[ x=\frac{24\pm8\sqrt{10}}{32} \]
\[ x=\frac{3\pm\sqrt{10}}{4} \]
Answer
\[
\boxed{x=\frac{3+\sqrt{10}}{4}\quad \text{or}\quad x=\frac{3-\sqrt{10}}{4}}
\]
The equation has two distinct real roots.