Quadratic equations

Equations in Quadratic Form

Some equations are not quadratic equations, but are in the same form, such as To solve equations such as that, you make a substitution, solve for the new variable, and then solve for the original variable

$\fs2x^4-5x^2+4=0\;\Rightarrow\;\;y=x^2$
$\fs2y^2-5y+4=0\;\Rightarrow\;y=\frac{5\pm\sqrt{(-5)^2-4\;1\;4}}{2\;1}\;\Rightarrow\;y=\frac{5\pm\sqrt{25-16}}{2}\;\Rightarrow\;y=\frac{5\pm3}{2}$

$\fs2y_1=\frac{5+3}{2}=\frac{8}{2}=4\;\;\;\;y_2=\frac{5-3}{2}=\frac{1}{2}=1$

These give

$\fs2x^2=y_1\;\;x^2=4\;\;\;x=\pm2\;\;\;x^2=y_2\;\;x^2=1\;\;\;x=\pm1\;\;\;$

Solve $\fs1x^4+14x^2+48=0$

No solution
One solution x=
Two solutions x₁= x₂=
Three solutions x₁= x₂= x₃=
Four solutions x₁= x₂= x₃= x₄=