Question

A equação 4x4-37x2+9=0 possui quatro raizes reais. Determine a soma dos quadrados dessas raizes.

Answer 1

$4x^4-37x^2+9=0$

reescrevendo
$x^2= a$

substituindo temos

$\boxed{4a^2-37a+9=0}$

resolvendo aplicando bhaskara

$a= \frac{-(-37)\pm \sqrt{(-37)^2-4*4*9} }{2*4} = \frac{37\pm \sqrt{1369-144} }{8} = \frac{37\pm \sqrt{1225} }{8} \\\\\\a= \frac{37\pm 35}{8} =\Bmatrix{a= \frac{37- 35}{8}= \frac{1}{2} \\\\a= \frac{-37+ 35}{8} =9 \end$

como a=x²
então

$\Bmatrix{x^2 = \frac{1}{2}\to \boxed{x=\pm \sqrt{ \frac{1}{2}}}\\\\\\x^2=9\to \boxed{x=\pm \sqrt{9} =\pm3} \end$

as raízes são
$\boxed{\boxed{[-3;-\sqrt{ \frac{1}{2}};\sqrt{ \frac{1}{2} } ;3]}}}$

a soma dos quadrados delas
$(-3)^2+\left(-\sqrt{ \frac{1}{2}}\right)^2+\left(\sqrt{ \frac{1}{2} }\right)^2+(3)^2\\\\=9+ \frac{1}{2} + \frac{1}{2}+9 \\\\=18+1= 19$