Factor the following trinomials. (Hint: Solve the equations first.)
x ²−2x −2
since solving by factoring assumes you know how to factor, let's complete the square instead.
x^2 - 2x - 2 = 0
x^2 - 2x = 2
x^2 - 2x + 1 = 2+1
(x-1)^2 = 3
x = 1±√3
so the factors are
(x - (1+√3))(x - (1-√3))
To factor the trinomial x^2 - 2x - 2, we need to find two binomials that, when multiplied together, give us the original trinomial.
To do this, we can follow these steps:
Step 1: Start by writing the trinomial in the form of (x + a)(x + b), where a and b are the unknown constants we're trying to find.
So, we have (x + a)(x + b) = x^2 - 2x - 2.
Step 2: Expand the binomial multiplication to get x^2 + (a+b)x + ab.
Now, we can see that a + b = -2, and ab = -2.
Step 3: We need to find two numbers a and b that satisfy the conditions from step 2.
Let's make a table of possible values for a and b:
a | b | a + b | ab
-----------------------------------------
-1 | -1 | -2 | 1
1 | -2 | -1 | -2
-2 | 1 | -1 | -2
2 | -1 | 1 | -2
From the table, we can see that a = 1 and b = -2 satisfy the conditions.
So, we can factor the trinomial as: (x + 1)(x - 2).
To factor a trinomial like x² - 2x - 2, we need to find two binomials whose product equals the given trinomial.
Here's how we can do it:
Step 1: Solve the equation x² - 2x - 2 = 0
To factor the trinomial, we need to find the values of x that make the equation equal to zero. This can be done by factoring or using the quadratic formula.
Using the quadratic formula:
The quadratic formula is x = (-b ± √(b² - 4ac)) / (2a)
In this case, a = 1, b = -2, and c = -2. Plugging these values into the quadratic formula, we get:
x = (-(-2) ± √((-2)² - 4(1)(-2))) / (2(1))
x = (2 ± √(4 + 8)) / 2
x = (2 ± √12) / 2
Simplifying further, we get:
x = (2 ± 2√3) / 2
x = 1 ± √3
Therefore, the solutions to the equation x² - 2x - 2 = 0 are x = 1 + √3 and x = 1 - √3.
Step 2: Use the solutions to factor the trinomial
Now that we have the solutions, we can express the quadratic equation as the product of two binomials. The binomials are formed by taking the opposite of each solution and putting them inside the parentheses.
x² - 2x - 2 = (x - (1 + √3))(x - (1 - √3))
Expanding this out, we get:
x² - 2x - 2 = (x - 1 - √3)(x - 1 + √3)
So, the factored form of the trinomial x² - 2x - 2 is (x - 1 - √3)(x - 1 + √3).