Equations that are Quadratic in Form

Example: Solve the equation.

$5x^{2/3}-6x^{1/3}+1=0$

Solution:

The equation is similar to a quadratic. It has 3 terms and one exponent is twice the other. Since the equation is quadratic in form, use substitution to solve the equation.

Use the following substitution to rewrite the equation

$u=x^{1/3}$

$u^2=x^{2/3}$

Original Equation	$5x^{2/3}-6x^{1/3}+1=0$
Substitute $u=x^{1/3}$ $u^2=x^{2/3}$	$5x^{2/3}-6x^{1/3}+1=0$ $5u^{2}-6u+1=0$
Solve the quadratic equation by factoring. 1) Factor the quadratic	$5u^{2}-6u+1=0$
Solve the quadratic equation by factoring. 2) Apply the zero product property	or
Solve the quadratic equation by factoring. 3) Solve each linear factor	or or or or or
Substitute again to bring back the original variable. Use the original substitution. $u=x^{1/3}$	or $x^{1/3}=1/5$ or $x^{1/3}=1$
Solve the equation with rational exponents. 1) Rewrite the rational exponents in radical form	$x^{1/3}=1/5$ or $x^{1/3}=1$ or
Solve the equation with rational exponents. 2) Cancel the cube root by cubing both sides. 3) Simplify	or $(root{3}{x})^3=(1/5)^3$ or $(root{3}{x})^3=(1)^3$ or