Determine Whether 2x² − 2√2x + 1 = 0 Has Real Roots and Find the Roots
Question
Determine whether the given quadratic equation has real roots and if so, find the roots:
\[ 2x^2-2\sqrt2x+1=0 \]Solution
\[ a=2,\quad b=-2\sqrt2,\quad c=1 \]
Find the discriminant:
\[ D=b^2-4ac \]
\[ D=(-2\sqrt2)^2-4(2)(1) \]
\[ D=8-8 \]
\[ D=0 \]
Since
\[ D=0 \]
the equation has two equal real roots.
\[ x=\frac{-b}{2a} \]
\[ x=\frac{2\sqrt2}{4} \]
\[ x=\frac{\sqrt2}{2} \]
Answer
\[
\boxed{x=\frac{\sqrt2}{2}}
\]
The equation has two equal real roots.