Video Transcript
Find the solution set of six 𝑥 squared minus 13𝑥 plus six equals zero in the set of real numbers.
There are numerous ways of approaching this problem. We will solve our equation by factoring. One way of factoring the quadratic 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 is to find two integers that have a product equal to 𝑎 multiplied by 𝑐 and a sum equal to 𝑏. In this question, the values of 𝑎, 𝑏, and 𝑐 are six, negative 13, and six, respectively. This means that we need to find two numbers with a product of 36 and a sum of negative 13. Multiplying two negative numbers gives a positive answer, and negative nine multiplied by negative four is 36. Negative nine and negative four sum to give us negative 13. This allows us to rewrite our quadratic by splitting the middle term into negative nine 𝑥 and negative four 𝑥. Our equation is six 𝑥 squared minus nine 𝑥 minus four 𝑥 plus six is equal to zero.
We can then factor the first two terms by finding the highest common factor and then repeat this process for the third and fourth terms. The highest common factor of six 𝑥 squared and negative nine 𝑥 is three 𝑥. Six 𝑥 squared minus nine 𝑥 can be rewritten as three 𝑥 multiplied by two 𝑥 minus three. Likewise, negative four 𝑥 plus six can be rewritten as negative two multiplied by two 𝑥 minus three. We notice that our parentheses are identical. We can therefore rewrite the quadratic as two 𝑥 minus three multiplied by three 𝑥 minus two.
As the product of our parentheses equals zero, either two 𝑥 minus three equals zero or three 𝑥 minus two equals zero. Adding three to both sides of our first equation and then dividing through by two gives us 𝑥 is equal to three over two. For our second equation, we can add two to both sides and then divide through by three. This gives us 𝑥 is equal to two-thirds. There are two possible solutions for the quadratic equation six 𝑥 squared minus 13𝑥 plus six equals zero. They are two-thirds and three-halves.
