Create solution video---Example 4: Find the roots of the quadratic equation 6x² - x - 2 = 0.
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Let's solve the quadratic equation 6x squared minus x minus 2 equals 0 using factoring. First, we identify the coefficients: a equals 6, b equals negative 1, and c equals negative 2. To factor this quadratic, we need to find two numbers that multiply to a times c, which is 6 times negative 2, equals negative 12, and add up to b, which is negative 1.
Now we need to find two numbers that multiply to negative 12 and add to negative 1. Let's check the factor pairs of negative 12. One times negative 12 equals negative 12, but 1 plus negative 12 equals negative 11. Two times negative 6 equals negative 12, but 2 plus negative 6 equals negative 4. Three times negative 4 equals negative 12, and 3 plus negative 4 equals negative 1. This is our answer! We need negative 4 and 3.
Now we rewrite the middle term negative x using our numbers negative 4 and 3. We replace negative x with negative 4x plus 3x, giving us 6x squared minus 4x plus 3x minus 2 equals 0. Next, we group the terms: the first group is 6x squared minus 4x, and the second group is 3x minus 2. Then we factor out the common factors from each group: 2x from the first group and 1 from the second group, giving us 2x times 3x minus 2 plus 1 times 3x minus 2 equals 0.
Now we can factor out the common binomial factor 3x minus 2, giving us 3x minus 2 times 2x plus 1 equals 0. To find the roots, we set each factor equal to zero. From 3x minus 2 equals 0, we get 3x equals 2, so x equals two-thirds. From 2x plus 1 equals 0, we get 2x equals negative 1, so x equals negative one-half. Therefore, the roots of the quadratic equation are x equals two-thirds and x equals negative one-half.
To summarize what we've learned: We successfully solved the quadratic equation 6x squared minus x minus 2 equals 0 using the factoring method. The key was finding the correct factor pair and using the grouping technique to reach our final answer of x equals two-thirds and x equals negative one-half.