Factor x^2+x-1

Problem

x2+x−1

Solution

1. Identify the coefficients of the quadratic expression a*x2+b*x+c Here, a=1 b=1 and c=−1

2. Check for rational roots by looking for two numbers that multiply to a*c=−1 and add to b=1 Since no such integers exist, the expression cannot be factored using simple integers.

3. Apply the quadratic formula to find the roots of the equation x2+x−1=0

x=(−b±√(,b2−4*a*c))/(2*a)

4. Substitute the values into the formula.

x=(−1±√(,1−4*(1)*(−1)))/(2*(1))

5. Simplify the expression under the radical.

x=(−1±√(,5))/2

6. Write the factors using the roots (r_1)=(−1+√(,5))/2 and (r_2)=(−1−√(,5))/2 in the form (x−(r_1))*(x−(r_2))

(x−(−1+√(,5))/2)*(x−(−1−√(,5))/2)

7. Simplify the signs within the factors.

(x+(1−√(,5))/2)*(x+(1+√(,5))/2)

Final Answer

x2+x−1=(x+(1−√(,5))/2)*(x+(1+√(,5))/2)

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