Factor a^2-2a-1

Problem

a2−2*a−1

Solution

1. Identify the type of expression. This is a quadratic trinomial of the form a*x2+b*x+c where a=1 b=−2 and c=−1

2. Check for integer factors. We look for two numbers that multiply to −1 and add to −2 Since the only integer factors of −1 are 1 and −1 which sum to 0 this trinomial cannot be factored using integers.

3. Apply the quadratic formula to find the roots of the equation a2−2*a−1=0

a=(−(−2)±√(,(−2)2−4*(1)*(−1)))/(2*(1))

4. Simplify the expression under the radical.

a=(2±√(,4+4))/2

a=(2±√(,8))/2

5. Simplify the square root and the fraction.

a=(2±2√(,2))/2

a=1±√(,2)

6. Write the factored form using the roots (r_1)=1+√(,2) and (r_2)=1−√(,2) The factored form is (a−(r_1))*(a−(r_2))

(a−(1+√(,2)))*(a−(1−√(,2)))

7. Distribute the negative signs inside the parentheses.

(a−1−√(,2))*(a−1+√(,2))

Final Answer

a2−2*a−1=(a−1−√(,2))*(a−1+√(,2))

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