Determine Whether √2x² + 7x + 5√2 = 0 Has Real Roots and Find the Roots
Question
Determine whether the given quadratic equation has real roots and if so, find the roots:
\[ \sqrt2x^2+7x+5\sqrt2=0 \]Solution
\[ a=\sqrt2,\quad b=7,\quad c=5\sqrt2 \]
Find the discriminant:
\[ D=b^2-4ac \]
\[ D=(7)^2-4(\sqrt2)(5\sqrt2) \]
\[ D=49-40 \]
\[ D=9 \]
Since
\[ D>0 \]
the equation has two distinct real roots.
\[ x=\frac{-b\pm\sqrt{D}}{2a} \]
\[ x=\frac{-7\pm3}{2\sqrt2} \]
\[ x=\frac{-4}{2\sqrt2} =-\frac{2}{\sqrt2} =-\sqrt2 \]
or
\[ x=\frac{-10}{2\sqrt2} =-\frac{5}{\sqrt2} =-\frac{5\sqrt2}{2} \]
Answer
\[
\boxed{x=-\sqrt2\quad \text{or}\quad x=-\frac{5\sqrt2}{2}}
\]
The equation has two distinct real roots.