Write the solution set of the equation x2+x−2=0 in roster form.
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Step-by-Step Solution
Step 1: Factor the quadratic equation
To find the solutions of the quadratic equation x2+x−2=0, we can factor the quadratic expression. We look for two numbers that multiply to −2 and add up to 1. These numbers are 2 and −1. So, we can rewrite the middle term x as 2x - x.
Step 2: Group terms and factor out common factors
Now, we group the terms and factor out the common factors from each pair. From the first two terms, x2+2x, we factor out x. From the last two terms, −x−2, we factor out −1. This gives us x(x+2)−1(x+2)=0.
Step 3: Factor out the common binomial
We can see that (x+2) is a common binomial factor in both terms. Factoring out (x+2) leaves us with (x−1)(x+2)=0.
Step 4: Solve for x
For the product of two factors to be zero, at least one of the factors must be zero. Therefore, we set each factor equal to zero and solve for x. This gives us x−1=0, which means x=1, and x+2=0, which means x=−2.
Step 5: Write the solution set in roster form
The solution set consists of all values of x that satisfy the equation. In roster form, we list these values within curly braces. So, the solution set is {1,−2}.