Solve by Factoring x^4-14x^2+45=0

Problem

x4−14*x2+45=0

Solution

1. Substitute a new variable to treat the equation as a quadratic. Let u=x2

u2−14*u+45=0

2. Factor the quadratic expression by finding two numbers that multiply to 45 and add to −14 These numbers are −9 and −5

(u−9)*(u−5)=0

3. Back-substitute x2 for u to return to the original variable.

(x2−9)*(x2−5)=0

4. Factor further using the difference of squares formula a2−b2=(a−b)*(a+b) for the first term, and treat 5 as (√(,5))2 for the second term.

(x−3)*(x+3)*(x−√(,5))*(x+√(,5))=0

5. Solve for x by setting each factor equal to zero using the zero product property.

x−3=0⇒x=3

x+3=0⇒x=−3

x−√(,5)=0⇒x=√(,5)

x+√(,5)=0⇒x=−√(,5)

Final Answer

x=±3,±√(,5)

Want more problems? Check here!