To factor the quadratic equation x^2 - 2x - 6 = 0, you need to find two numbers whose product is equal to the product of the coefficient of x^2 (which is 1) and the constant term (which is -6), and whose sum is equal to the coefficient of x (which is -2). Let's proceed with the factoring.
Step 1: Write down the equation: x^2 - 2x - 6 = 0
Step 2: Look at the coefficient of x^2, which is 1, and the constant term, which is -6.
Step 3: Find two numbers whose product is -6 and whose sum is -2. In this case, the numbers are -3 and 2, as (-3) * (2) = -6 and (-3) + (2) = -1.
Step 4: Rewrite the middle term (-2x) in terms of these two numbers. Replace -2x with -3x + 2x, as -3x + 2x is equivalent to -2x. The equation becomes:
x^2 - 3x + 2x - 6 = 0
Step 5: Group the terms in pairs and factor by grouping. Factor out the common factors from each pair separately. The equation becomes:
(x^2 - 3x) + (2x - 6) = 0
Now factor out the greatest common factor from each pair:
x(x - 3) + 2(x - 3) = 0
Step 6: Combine the two terms with the same factor:
(x + 2)(x - 3) = 0
Step 7: Set each factor equal to zero and solve for x:
x + 2 = 0 or x - 3 = 0
Solving each equation separately:
For x + 2 = 0, subtract 2 from both sides:
x = -2
For x - 3 = 0, add 3 to both sides:
x = 3
Step 8: Check your solutions by substituting x = -2 and x = 3 back into the original equation:
For x = -2:
(-2)^2 - 2(-2) - 6 = 0
4 + 4 - 6 = 0
8 - 6 = 0
2 = 0 (False)
For x = 3:
(3)^2 - 2(3) - 6 = 0
9 - 6 - 6 = 0
3 - 6 = 0
-3 = 0 (False)
Therefore, there are no real solutions to x^2 - 2x - 6 = 0.