Solve the Quadratic Equation by Factorization: 9x² − 6b²x − (a⁴ − b⁴) = 0

Question: Solve the following quadratic equation by factorization:

$$ 9x^2-6b^2x-(a^4-b^4)=0 $$

Solution

$$ 9x^2-6b^2x-a^4+b^4=0 $$

$$ 9x^2-6b^2x+b^4-a^4=0 $$

$$ (3x-b^2)^2-a^4=0 $$

$$ (3x-b^2-a^2)(3x-b^2+a^2)=0 $$

Either

$$ 3x-b^2-a^2=0 $$

$$ x=\frac{a^2+b^2}{3} $$

or

$$ 3x-b^2+a^2=0 $$

$$ x=\frac{b^2-a^2}{3} $$

Hence,

$$ \boxed{x=\frac{a^2+b^2}{3},\ \frac{b^2-a^2}{3}} $$